physics of comets 3rd ed. - k. swamy

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WORLD SCIENTIFIC SERIES IN ASTRONOMY AND ASTROPHYSICS

Editor: Jayant V. NarlikarInter-University Centre for Astronomy and Astrophysics, Pune, India

Published:

Volume 1: Lectures on Cosmology and Action at a Distance ElectrodynamicsF. Hoyle and J. V. Narlikar

Volume 2: Physics of Comets (2nd Ed.)K. S. Krishna Swamy

Volume 3: Catastrophes and Comets*V. Clube and B. Napier

Volume 4: From Black Clouds to Black Holes (2nd Ed.)J. V. Narlikar

Volume 5: Solar and Interplanetary DisturbancesS. K. Alurkar

Volume 6: Fundamentals of Solar AstronomyA. Bhatnagar and W. Livingston

Volume 7: Dust in the Universe: Similarities and DifferencesK. S. Krishna Swamy

Volume 8: An Invitation to AstrophysicsT. Padmanabhan

Volume 9: Stardust from Meteorites: An Introduction to Presolar GrainsM. Lugaro

Volume 10: Rotation and Accretion Powered PulsarsP. Ghosh

Volume 11: Find a Hotter Place!: A History of Nuclear AstrophysicsL. M. Celnikier

Volume 12: Physics of Comets (3rd Edition)K. S. Krishna Swamy

*Publication cancelled.

Lakshmi - Physics of Comets (3rd Ed).pmd 7/12/2010, 2:46 PM2

N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I

World Scientific

K S Krishna Swamyretired from the Tata Institute of Fundamental Research, Mumbai, India

World Scientific Series in Astronomy and Astrophysics – Vol. 12

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PHYSICS OF COMETS (3rd Edition)World Scientific Series in Astronomy and Astrophysics — Vol. 12

Lakshmi - Physics of Comets (3rd Ed).pmd 7/12/2010, 2:46 PM1

July 12, 2010 9:59 World Scientific Book - 9in x 6in comets

Foreword

Only in recent years has the true nature of comets become evident.

With this knowledge has come the realization that comets are almost cer-

tainly debris left over from the building of the outermost planets. This

material must be typical of the gas and dust in the interstellar cloud from

which the Sun and planetary system evolved. Thus, comets appear to offer

us an opportunity to study primitive matter involved in the origin of the

solar system, material that has been stored in deep freeze for 4.6 × 109

years. At the same time, the comets provide a link with interstellar solids,

of which they are probably largely composed. The physical and chemical

study of comets has now replaced classical celestial mechanics as the major

focus of observational and theoretical research. Familiar areas of astronom-

ical spectroscopy are approached from a somewhat unusual point of view in

cometary studies, while comets provide an entirely new laboratory for an in-

creased understanding of magneto-hydrodynamics. On the other extreme,

the physics of amorphous ices and other solids can be applied to probe the

internal structure, origin and activity of the comet nuclei. In summation,

the study of comets has now become a manifold discipline with fascinating

potential, particularly as the space age provides new observational input

over the entire electromagnetic spectrum, coupled with the expectation of

direct in situ studies and even the eventual return of samples from cometary

nuclei. The marked progress in this direction is represented by the new re-

sults described in this book’s revised (second) edition. The several missions

to Halley’s comet have filled in many lacunae of comet knowledge.

Fred L. Whipple

v

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Preface

We are in an exciting era of cometary science. This has come about due

to space mission to comets, launch of several satellites and ground based

studies. The first major space adventure was in 1986 when six spacecraft

passed through Comet Halley as close as 500 km from the nucleus. More

recently, the far more adventures venture of the Deep Impact Mission to

Comet Tempel 1 and Stardust Mission to Comet Wild 2 made it possible for

the first time to look at the material in the deeper layers of the cometary

nuclei and to study in the laboratory the collected cometary dust. All

these studies have given rise to exciting and unexpected results which have

raised some fundamental issues about comets and in turn on the solar

system itself. For example, the detection of X-rays from comets was a

real challenge in its understanding. The observed diversity in chemical

composition among comets has direct bearing on the formation conditions

in the solar nebula. The mineralogy of dust in IDPs, comets, meteorites,

asteroids and circumstellar shell of stars are quite similar showing generic

relationship between them.

In view of these exciting developments it is an appropriate time to revise

and update the second edition of the book “Physics of Comets” published

in 1997. Therefore extensive revision has been carried out in most of the

chapters. Emphasis has been to include all the new results with the hope

that the flavour of excitement may be conveyed to the reader. The general

arrangement, purpose and level of the updated and revised edition of the

book remains the same as that of second edition.

I am especially grateful to my colleagues, H.M. Antia who helped me in

various ways and especially in the preparation of figures and S. Ramadurai

for a thorough reading of the manuscript. I would also like to express my

gratitude to my family and parents for their support and encouragement.

vii


viii Physics of Comets

I would like to thank all the publishers and authors who have given per-

mission for the reproduction of figures and tables. The help of Mr. Vivek

Vengurlekar in the preparation of the manuscript is highly appreciated. My

interaction with several astronomers have been very helpful and especially,

the cometary experts like, M.F.A’Hearn, P.D. Feldman, W.F. Huebner,

M.J. Mumma, C.R.O’Dell and J. Watanabe.

K S Krishna Swamy

204, Sigma Tower

Plot 32, Gorai Road

Borivali West

Mumbai 400 091, India


Contents

Foreword v

Preface vii

1. General Introduction 1

1.1. Historical Perspective . . . . . . . . . . . . . . . . . . . 1

1.2. Discovery . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3. Appearance . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4. Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5. Importance . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6. Brightness . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.7. Main Characteristics . . . . . . . . . . . . . . . . . . . 13

1.8. Spacecraft Encounters with Comets . . . . . . . . . . . 19

1.9. An Overall View . . . . . . . . . . . . . . . . . . . . . . 23

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2. Dynamics 33

2.1. Orbital Elements . . . . . . . . . . . . . . . . . . . . . . 33

2.2. Orbit in Space . . . . . . . . . . . . . . . . . . . . . . . 35

2.2.1. Relevant equations . . . . . . . . . . . . . . . . 35

2.2.2. Orbital elements from position and velocity . . 40

2.2.3. Orbital elements from observations . . . . . . 41

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

ix


x Physics of Comets

3. Physical Aspects 47

3.1. Black Body Radiation . . . . . . . . . . . . . . . . . . . 47

3.2. Perfect Gas Law . . . . . . . . . . . . . . . . . . . . . . 49

3.3. Dissociative Equilibrium . . . . . . . . . . . . . . . . . 49

3.4. Doppler Shift . . . . . . . . . . . . . . . . . . . . . . . . 50

3.5. Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5.1. Atomic spectroscopy . . . . . . . . . . . . . . . 51

3.5.2. Molecular spectroscopy . . . . . . . . . . . . . 54

3.5.3. Chemical subgroups . . . . . . . . . . . . . . . 57

3.6. Isotopic Effect . . . . . . . . . . . . . . . . . . . . . . . 58

3.7. Franck-Condon Factors . . . . . . . . . . . . . . . . . . 59

3.8. Intensity of Emitted Lines . . . . . . . . . . . . . . . . 61

3.9. Boltzmann Distribution . . . . . . . . . . . . . . . . . . 64

3.10. Λ-Doubling . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.11. Photochemistry of Water . . . . . . . . . . . . . . . . . 65

3.12. Silicate . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.13. Annealing . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.14. Carbon . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.15. Solar Radiation . . . . . . . . . . . . . . . . . . . . . . 70

3.16. Solar Wind . . . . . . . . . . . . . . . . . . . . . . . . . 72

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4. Spectra 75

4.1. Main Characteristics . . . . . . . . . . . . . . . . . . . 76

4.2. Forbidden Transitions . . . . . . . . . . . . . . . . . . . 90

4.3. Line–to–Continuum Ratio . . . . . . . . . . . . . . . . 92

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5. Spectra of Coma 95

5.1. Fluorescence Process . . . . . . . . . . . . . . . . . . . 95

5.1.1. Rotational structure . . . . . . . . . . . . . . . 98

5.1.2. Vibrational structure . . . . . . . . . . . . . . 101

5.1.3. Comparison with observations . . . . . . . . . 102

5.1.4. Case of C2 molecule . . . . . . . . . . . . . . . 108

5.1.5. Prompt emission lines of OH . . . . . . . . . . 114

5.1.6. Molecules other than diatomic . . . . . . . . . 118


Contents xi

5.1.7. OH radio lines . . . . . . . . . . . . . . . . . . 118

5.1.8. Oxygen lines . . . . . . . . . . . . . . . . . . . 121

5.1.9. Forbidden transitions . . . . . . . . . . . . . . 121

5.1.10. Molecular band polarization . . . . . . . . . . 123

5.2. Excitation Temperature . . . . . . . . . . . . . . . . . . 125

5.2.1. Rotational temperature . . . . . . . . . . . . . 125

5.2.2. Vibrational temperature . . . . . . . . . . . . 128

5.3. Abundances of Heavy Elements . . . . . . . . . . . . . 128

5.4. Isotopic Abundances . . . . . . . . . . . . . . . . . . . . 130

5.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6. Gas Production Rates in Coma 141

6.1. Theoretical Models . . . . . . . . . . . . . . . . . . . . 142

6.1.1. From the total luminosity . . . . . . . . . . . . 142

6.1.2. From surface brightness distribution . . . . . . 144

6.1.3. From number densities . . . . . . . . . . . . . 149

6.1.4. Semi-empirical photometric theory . . . . . . . 151

6.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.2.1. OH and H . . . . . . . . . . . . . . . . . . . . 152

6.2.2. H2O, H2 . . . . . . . . . . . . . . . . . . . . . 158

6.2.3. CN, C2, C3, NH . . . . . . . . . . . . . . . . . 162

6.2.4. CH, NH2 . . . . . . . . . . . . . . . . . . . . . 166

6.2.5. CO, CO2 . . . . . . . . . . . . . . . . . . . . . 167

6.2.6. CS, S2 . . . . . . . . . . . . . . . . . . . . . . 169

6.2.7. Ions . . . . . . . . . . . . . . . . . . . . . . . . 169

6.2.8. Complex molecules . . . . . . . . . . . . . . . 171

6.2.9. O, C, N, S . . . . . . . . . . . . . . . . . . . . 176

6.3. Analysis of Hydrogen Observations . . . . . . . . . . . 183

6.3.1. Analysis of Lyman α measurements . . . . . . 183

6.3.2. Analysis of Hα observations . . . . . . . . . . 189

6.4. Related Studies . . . . . . . . . . . . . . . . . . . . . . 191

6.4.1. Gas-phase chemistry in the coma . . . . . . . 191

6.4.2. In situ mass spectrometer for ions . . . . . . . 194

6.4.3. Temperature and velocity of the coma gas . . 196

6.5. Parent Molecules . . . . . . . . . . . . . . . . . . . . . . 202

6.6. Chemical Diversity . . . . . . . . . . . . . . . . . . . . 206

6.7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 207


xii Physics of Comets

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

7. Dust Tails 213

7.1. Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 213

7.2. Anti-tail . . . . . . . . . . . . . . . . . . . . . . . . . . 220

7.3. Dust Trails . . . . . . . . . . . . . . . . . . . . . . . . . 222

7.4. Sodium Gas Tails . . . . . . . . . . . . . . . . . . . . . 222

7.5. Dust features . . . . . . . . . . . . . . . . . . . . . . . . 223

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

8. Light Scattering Theory 227

8.1. Mie Scattering Theory . . . . . . . . . . . . . . . . . . 228

8.1.1. Efficiency factors . . . . . . . . . . . . . . . . . 228

8.1.2. Albedo . . . . . . . . . . . . . . . . . . . . . . 230

8.1.3. Scattered intensity . . . . . . . . . . . . . . . . 231

8.1.4. Polarization . . . . . . . . . . . . . . . . . . . 232

8.2. Approximate Expressions . . . . . . . . . . . . . . . . . 233

8.3. Computation of Cross Sections . . . . . . . . . . . . . . 233

8.4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

8.5. Particles of Other Types . . . . . . . . . . . . . . . . . 238

8.6. Optical Constants . . . . . . . . . . . . . . . . . . . . . 243

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

9. The Nature of Dust Particles 249

9.1. Visible Continuum . . . . . . . . . . . . . . . . . . . . . 249

9.1.1. Albedo . . . . . . . . . . . . . . . . . . . . . . 255

9.1.2. Phase function . . . . . . . . . . . . . . . . . . 256

9.1.3. Dust production rate from continuum . . . . . 257

9.1.4. Dust production from A(θ)fρ . . . . . . . . . 259

9.2. Polarization . . . . . . . . . . . . . . . . . . . . . . . . 261

9.2.1. Linear polarization . . . . . . . . . . . . . . . 261

9.2.2. Circular polarization . . . . . . . . . . . . . . 266

9.3. Grain Sizes . . . . . . . . . . . . . . . . . . . . . . . . . 268

9.4. Infrared Measurements . . . . . . . . . . . . . . . . . . 269

9.4.1. Dust production from infrared observations . . 272


Contents xiii

9.4.2. Anti-tail . . . . . . . . . . . . . . . . . . . . . 276

9.5. Spectral Feature . . . . . . . . . . . . . . . . . . . . . . 278

9.5.1. Silicate signature . . . . . . . . . . . . . . . . . 279

9.5.2. Mineralogy of dust particles . . . . . . . . . . 281

9.5.3. The C-H stretch feature . . . . . . . . . . . . . 288

9.5.4. Ice signature . . . . . . . . . . . . . . . . . . . 289

9.6. Properties Derived from Direct Measurements . . . . . 290

9.7. Radiation Pressure Effects . . . . . . . . . . . . . . . . 294

9.8. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 296

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

10. Ion Tails 303

10.1. Evidence for the Solar Wind . . . . . . . . . . . . . . . 303

10.2. Dynamical Aberration . . . . . . . . . . . . . . . . . . . 304

10.3. Theoretical Considerations . . . . . . . . . . . . . . . . 308

10.3.1. Comparison with observations . . . . . . . . . 314

10.4. Instabilities and Waves . . . . . . . . . . . . . . . . . . 318

10.5. Acceleration of Cometary Ions . . . . . . . . . . . . . . 321

10.6. Large Scale Structures . . . . . . . . . . . . . . . . . . 323

10.6.1. Tail rays or streamers . . . . . . . . . . . . . . 324

10.6.2. Knots or condensations . . . . . . . . . . . . . 325

10.6.3. Oscillatory structure . . . . . . . . . . . . . . . 325

10.6.4. “Swan-like” feature . . . . . . . . . . . . . . . 326

10.6.5. Bend in the tail . . . . . . . . . . . . . . . . . 326

10.6.6. Disconnection events . . . . . . . . . . . . . . 328

10.7. X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

10.8. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 335

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

11. Nucleus 339

11.1. Morphology . . . . . . . . . . . . . . . . . . . . . . . . 339

11.2. Theory of Vapourization . . . . . . . . . . . . . . . . . 340

11.3. Outbursts . . . . . . . . . . . . . . . . . . . . . . . . . . 348

11.4. Albedo and Radius . . . . . . . . . . . . . . . . . . . . 350

11.5. Mass, Density and Surface Gravity . . . . . . . . . . . . 353

11.6. Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 355


xiv Physics of Comets

11.7. Nucleus Composition . . . . . . . . . . . . . . . . . . . 359

11.8. Mass Loss . . . . . . . . . . . . . . . . . . . . . . . . . 360

11.9. Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 361

11.10. Non-gravitational Forces . . . . . . . . . . . . . . . . . 365

11.11. Ortho to Para Ratio of Molecules . . . . . . . . . . . . 370

11.12. Binary Systems . . . . . . . . . . . . . . . . . . . . . . 375

11.13. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 376

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

12. Origin 381

12.1. Evidence for the Oort Cloud . . . . . . . . . . . . . . . 381

12.2. Evolution and Properties of Oort Cloud . . . . . . . . . 384

12.2.1. Short period comets . . . . . . . . . . . . . . . 389

12.3. Origin of the Oort Cloud . . . . . . . . . . . . . . . . . 391

12.4. Taxonomy . . . . . . . . . . . . . . . . . . . . . . . . . 396

12.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 398

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

13. Relation to Other Solar System Studies 401

13.1. Asteroids . . . . . . . . . . . . . . . . . . . . . . . . . . 401

13.2. Meteorites . . . . . . . . . . . . . . . . . . . . . . . . . 406

13.3. Meteor Streams . . . . . . . . . . . . . . . . . . . . . . 411

13.4. Particles Collected at High Altitudes . . . . . . . . . . 416

13.5. Primordial Material . . . . . . . . . . . . . . . . . . . . 419

13.6. Chemical Evolution . . . . . . . . . . . . . . . . . . . . 420

13.7. Terrestrial Water . . . . . . . . . . . . . . . . . . . . . 423

13.8. Impact of Outside Bodies . . . . . . . . . . . . . . . . . 424

13.9. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 426

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

14. Problems and Prospects 435

14.1. Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . 435

14.2. Future studies . . . . . . . . . . . . . . . . . . . . . . . 437

Index 441


CHAPTER 1

General Introduction

1.1. Historical Perspective

Among the various objects of the solar system, comets have attracted

and fascinated the common man to a large extent for the last two thousand

years or so. This information comes from the ancient records of paintings or

drawings of comets on caves, clothes, etc. as well as from the observations

of early writers. It was not until the sixteenth century that comets were

demonstrated to be celestial objects. This came from the work of Tycho

Brahe who observed the bright comet of 1577 AD with accurate instruments

and from various locations in Europe. This really revolutionized the ideas

about comets and from then on, observers took a serious view of comets

and started making position measurements.

The complete credit for the discovery that comets are part of the so-

lar system goes to Edmond Halley. Halley, using Newtonian mechanics,

showed that the comets which had appeared in 1531, 1607 and 1682 are

the one and the same with a period of about 7512 years. He also noticed

that the time interval between the successive perihelion passages was not

the same. He concluded rightly that this could be due to the perturbation

of the cometary orbit produced by the planets Jupiter and Saturn. Fol-

lowing these successes, he predicted that the same comet would return in

1758. As predicted, the comet did appear in 1758, though Halley, dead by

then, was not there to witness the glorious triumph of his prediction. This

comet is therefore named after him. In recent years this comet has been

traced backwards through many centuries by several investigators through

orbit calculations. Through this work, it has been possible to identify every

appearance of the comet as shown by ancient records until about 240 BC.

So far, it seems to have made about twenty-eight appearances.

1


2 Physics of Comets

Since early times, the appearance of a comet has been associated

with disasters, calamities, tragedies and so on. One beneficial result of

such wrong notions and ideas is that the appearances of most comets are

recorded. These observations have proved very valuable to modern as-

tronomers. Although there were many bright comets which have been

recorded since early times, somehow the Comet Halley seems to have at-

tracted much more attention than the others. The comet which has been

depicted in the Bayeux tapestry is the Comet Halley which appeared in

1066 AD (Fig. 1.1). The comet in the tapestry can be seen to hover above

Fig. 1.1 A portion of the Bayeux tapestry showing the 1066 apparition of Comet Halley.

The tapestry depicts the people pointing at the comet with fear for its effect on King

Harold of England (Report of the Science Working Group. The International Halley,Watch, NASA, July 1980).

the English King Harold who is being told of the bad omen. Such types

of association of Comet Halley with the occurrence of bad things on Earth

have also been made for many other apparitions. The last apparition of

Comet Halley in 1910 drew wide publicity (Fig. 1.2). It was very bright

and enormous in extent. Actually the Earth passed through the tenuous

gas of this comet’s tail.

There was also the fear that a comet might collide with the Earth and

bring disastrous consequences. There are several indirect observational ev-

idences, such as the presence of Cretaceous-Tertiary boundary, Tunguska

event and so on, which show that such events must have happened on the

Earth in the past. The collision of Comet Shoemaker-Levy 9 with Jupiter

during July 1994 has given supporting evidence, although the probability

of such an event happening on the Earth is very small. With the passage of

time some of the fears have been erased from the people’s minds. Today the

appearance of a bright comet in the sky like that of the Comet Ikeya-Seki of


General Introduction 3

Fig. 1.2 Some of the newspaper headlines which appeared in The New York Times

during Comet Halley’s appearance in 1910 (Report of the Science Working Group, loc,

cit.).

1965, Comet West in 1975, Comet Hale-Bopp in 1995 or the Comet Hyaku-

take in 1996 is welcomed both by the scientists and the public at large.

Scientists look forward to observing and studying these objects and under-

standing their nature. The public look forward to viewing a spectacular

and colourful event in the sky.

During early times, comets were studied more from the point of their

dynamics. This was made possible through the efforts of many pioneers in

celestial mechanics. These extensive dynamical studies of various comets

have shown, for the first time, the existence of some important physical

effects like the presence of non-gravitational forces in comets. With the

passage of time, cometary research has evolved from the study of dynamics


4 Physics of Comets

to the study of these objects per se. Specifically, in the last three or four

decades, emphasis has been laid more in understanding the origin, physics

and chemistry of these objects. The presence of complex organic molecules

including molecules of biological interest in comets, which may have some

relation to the existence of life on Earth, has interested biologists too, in

the study of comets.

1.2. Discovery

Many comets are discovered by amateur astronomers who just scan the

sky with a low-power telescope. They are called ‘comet seekers’. The

comets are usually named after their discoverers. If two or even more

observers find the same comet nearly simultaneously, all the names are

attached to that comet. For example, the Comet Ikeya-Seki (1965 VIII)

which was visible to the naked eye in 1965, was discovered by two Japanese

amateurs, Ikeya and Seki. However, not all comets are found by amateurs.

Many are being discovered nowadays by astronomers in their photographic

plates taken from some other scientific study. A typical example of this class

is Comet Kohoutek (1973 XII) discovered in 1973. Comets discovered by

them are fainter as the professional astronomers have better instruments.

Several comets have also been discovered through satellites, such as Solar

Heliospheric Observatory (SOHO). In addition to the names of the dis-

coverers, comets are also assigned temporary designations, indicating the

year of their discovery followed by a small letter denoting the order of their

discovery in that year. For example, the first two comets found in the

year say 1968, are designated as 1968a and 1968b respectively. Later on

when the orbits of all the comets discovered in that particular year are well

determined, permanent designations are given. This consists of the year

in which the comet passed nearest to the Sun i.e., perihelion, followed by

a Roman numeral which indicates the order of perihelion passage during

that year. Hence, the comets mentioned above will be given the permanent

designations 1968I and 1968II and so on. If the comet is periodic, one also

attaches P to the name of the comet. Thus the periodic Comet Encke is

written as P/Encke.

However, in this system of naming of comets, there is the difficulty

sometimes in deciding whether a particular object is a comet or a minor

planet. e.g. 2060 Chiron, 1990 UL3=1990P=1990XVI. In addition, con-

fusion can creep in when the better orbit of the comet becomes available



which could change the prediction of the order of perihelian passage by

even a year. In view of this, a change in the cometary designation is being

followed effective January 1, 1995. In the new system a year is divided

into fortnights assigning successively the Roman alphabet A to Z (leaving

out the letters I and Z). The designation of the comets consists of year

followed by the alphabet representing the fortnight of observation and the

order in which it was discovered. As for example, the third comet reported

as discovered during the second half of February 1995 would be designated

as 1995D3. If there is an indication of the nature of the object it could

be expressed by preceding the designation with C/(for comet), P/(as now,

for periodic comet), etc. Routine recoveries of periodic comets will not in

future receive additional designations. The new scheme also proposes to re-

tain the tradition of naming comets after their discoverers such as C/1996

B2(Hyakutake).

1.3. Appearance

Comets spend almost all their time at great distances from the Sun. The

cometary activity starts showing up only when it approaches the Sun. At

far-off distances from the Sun, it appears as a faint fuzzy patch of light. The

fuzzy patch of light is a cloud of gas and dust called coma. The coma grows

in size and brightens as it nears the Sun. In addition to the brightening

of the coma, the tail starts developing and reaches its maximum extent at

about the closest approach to the Sun. After its perihelion passage, the re-

verse process takes place in the sense that the comet starts fading away as it

recedes from the Sun. These effects can be seen clearly in the time sequence

photographs of Comet Halley Halley, as shown in Fig. 1.3. Cometary ac-

tivity is therefore transient in nature. These observations clearly show that

the material composed of gas and dust must have come from a central com-

pact solid source called the nucleus of comet. The diameter of the nucleus

is extremely small and is estimated to be about 1 to 10 km. This size is so

small that it appears as a point source and cannot be resolved even with

the largest telescope. On the other hand, the diameter of the coma is much

larger and lies in the range of about 104 to 105 km. The nucleus and the

coma forms the head of a comet. The most characteristic feature of a comet

is the tail which may extend up to about 107 to 108 km.


6 Physics of Comets

Fig. 1.3 Shows the time sequence photographs of Comet Halley in 1910 which brings

out the transient nature of comets (Report of the Science Working Group, loc. cit.).

1.4. Statistics

The rate of discovery of comets has increased steadily since the begin-

ning of the century. In recent times more and more fainter comets are also

being discovered due to vast improvement in the observational techniques

and also due to several systematic search programmes. On an average,

around 25 to 30 comets are seen every year. Of these, the newly discovered

comets are around 12 to 15 per year and the recoverable ones are around

13 to 15 per year. Comets are classified based on their orbital periods. The

general convention which has been followed is that short-period comets are

those which have periods (P) less than 200 years. Those which have peri-

ods greater than 200 years are called long-period comets. The short period

comets are further subdivided into two classes. Comets with periods < 20

years are called Jupiter-family comets (JFCs) and cluster around Jupiter



(semi major axis ≤ 7.4 AU). Those comets whose periods is in the range

20 < P < 200 years are called Halley- type comets (HTCs) with Comet

Halley as prototype. The semi major axis lies in the range of around 7.4 to

34.2 AU. Halley - type comets may be considered as short-period extension

of long-period comets. Among the newly discovered comets, about 8–10

are long period comets and 5 are short period ones. Most of these comets

are generally faint. The bright comets and, in particular, the sun-grazing

comets occur occasionally. Comets have been seen as close as 0.01 AU from

the Earth. Comets have also been classified as ‘old’ and ‘new’ based purely

on their orbital characteristics. Comets which have made several perihelion

passages around the Sun are generally termed ‘old’ and those which are

entering for the first time are called ‘new’. If the direction of motion of the

comet is the same as that of the Earth’s motion, in its orbit, it is said to

have a direct orbit. If they are in opposite directions, the comet is said to

have a retrograde orbit.

More than 1000 Comets are known. The ratio of long-period to short-

period comets is ≈ 5 : 1. For Jupiter-family comets, the peak in the orbital

period occurs around 7 to 8 years. The median orbital period of Halley-

type comets is around 70.5 years. But in short-period comets, the period

of a few years has also been seen as in the case of Comet Encke which has a

period of 3.3 years. Among the long period comets, most of them seem to

have parabolic and osculating elliptical orbits. It is of interest to know how

far the comets reach away from the Sun in their orbit, which is called the

aphelion distance. Figure 1.4 shows a histogram of the number of comets

versus aphelion distance. A peak in the distribution occurs around 5 AU,

which corresponds roughly to the distance of Jupiter which are the Jupiter-

family comets. The general pattern of the histogram remain roughly the

same with the addition of more comets. On the other hand, the long-period

comets seem to peak around 4×104 AU (Fig. 12.1). Those with a < 104 AU

are often called ‘returning’ comets and those with a > 104 AU are called

dynamically ‘new’ comets. It is also found that for most of the comets the

closest approach to the Sun, called the perihelion distance, is around 0.6 to

1.5 AU. The distribution of inclination to the ecliptic plane of short and long

period comets are striking. The short period comets have inclinations < 20

to 30 and the median inclination is around 11. The median inclination

for Halley-type comets is around 64. However, for long-period comets, the

inclinations are randomly distributed. They also approach the Sun more

or less isotropically. The short-period comets generally have direct orbits

while long-period comets have both direct and retrograde oribits.


8 Physics of Comets

Fig. 1.4 Shows the distribution of the number of comets as a function of the apheliondistance for short period comets. The peak around Jupiter’s distance can clearly be

seen. Smaller peaks at the distances of other planets also appear to be present (Bailey,

M.E., Clube, S.V.M. and Napier, W.M. 1990. The Origin of Comets, Pergamon Press).

There is a group of sun-grazing comets called Kreutz family of comets.

Several hundred such comets have been discovered by satellies such as

SOHO. This is because these comets are hardly visible from the earth due

to their intrinsic faintness and also because of their peculiar orbit charac-

teristics. They all appear to have the same orbital elements. The orbit of

these comets indicate an aphelion distance of around 120–200 AU from the

sun. This corresponds to a time period of revolution of around 500–1000

years. Their inclination peak is around 145. This indicates that these sun-

grazing comets are likely to have resulted from the break up of a very big

comet into large number of parts, sometime in the past. These sun-grazing

comets disappear during their approach to the sun. An example of such a

comet is shown in Fig. 1.10.



Age of a comet is generally measured by the reciprocal of the semi major

axis i.e. a−1 (AU−1). New comets coming from the Oort cloud for the first

time have a > 104 AU or (1/a) < 100×10−6 AU−1. With successive passges

the orbit shrinks gradually primarily due to planetary perturbations and

hence the value of (1/a) becomes larger and larger. Consequently, the

increasing value of (1/a) corresponds to increasing time with respect to the

first approach to the solar system. In other words, (1/a) gives a measure

of the comet’s age.

1.5. Importance

The study of comets is important from several points of view. Cometary

activity arises basically from the solar heating of the nucleus, releasing the

gas and dust, which finally are lost into the solar system. The time which

the comet spends near the Sun is a very small fraction of its total period. So

only a thin layer of the material of the nucleus is ablated at every perihelion

passage and nothing much would have happened in the inner regions of the

nucleus of a comet. The inner core of the comet may thus represent the

composition of the original material at the time of its formation. Therefore

it is hoped that a systematic study of the material of the nucleus of comets

can give information with regard to the nature of the material present at

the early phase of the solar nebula, 4 to 5 billion years ago, even before

the formation of the Earth and the solar system. The isotopic anomalies

seen in the cometary dust particles indicate the presence of presolar grains.

i.e. they refer to grains produced in the circumstellar shell of stars. This

indicate the presence of interstellar grains in comets. The highly complex

molecules and organic compounds seen in comets can finally find their way

on to the Earth. These might have played a key role in the complex scenario

of chemical evolution finally leading to life on the Earth. The tail pointing

away from the Sun arise primarily due to the interaction of the dust and gas

of the cometary material with the solar radiation and solar wind. Therefore

the study of cometary tails may throw light on the physical conditions of the

interplanetary medium as well as of the solar wind and the solar activity.

It is also of great interest from the point of view of Plasma Physics for

the study of interactions, generation of instabilities and waves and so on,

many of which cannot be produced under the laboratory conditions. In

fact, the existence of the solar wind, i.e. the flow of high velocity charged

particles from the Sun, was predicted by Biermann in the 1950s from the


10 Physics of Comets

study of the ion tails of comets. After a large number of revolutions around

the Sun, the cometary activity may die out completely leading finally to

a residual solid nucleus, which possibly may lead to an asteroid. Comets

are also believed to be the sources of meteors and interplanetary dust. It

is generally believed that the origin of comets is intimately related to the

origin of the solar system, a problem of great current interest. Therefore,

the study of comets can provide clues which may help in understanding the

origin and evolution of the solar system. There is also generic relationship

between comets with the interplanetary dust particles, meteorites, asteroids

and interstellar matter. In addition to these possible interrelationships, the

comets themselves are interesting objects to study, as their nature and

origin are still not well understood.

1.6. Brightness

One of the uncertain facts about a comet is its brightness. The comet

shines mostly due to the reflected sunlight at far-off distances from the Sun.

The brightness depends upon three factors: (i) the distance r from the Sun

to the comet: (ii) the nature of the comet and (iii) the distance ∆ from the

comet to the earth. The brightness depends upon the nature of the comet

as it is the one which is producing the observed radiation.

The expected brightness of a comet I, can be written as

I =Io

r2∆2φ(α) (1.1)

where φ(α) is the appropriate phase function which is not important for

the total brightness, Io is the constant of proportionality, usually taken to

be the brightness of the comet at r = ∆ = 1 AU. It has been found that

the brightness of comets rarely follow a simple relation of the above type.

Figure 1.5 shows results for a few comets. Mostly the power of r is greater

than 2. One usually writes a modified form of Eq. (1.1) as

I =Io

rn∆2. (1.2)

The above equation can be written in terms of magnitudes as

m = m0 + 5 log ∆ + 2.5n log r (1.3)

where m refers to the total apparent magnitude and m0 the absolute magni-

tude formally corresponding to r = ∆ = 1 AU. The study of a large number

of comets has given a pretty good idea as to the variation of brightness with



Fig. 1.5 The magnitude of the comet reduced to a standard distance of 1 AU from theEarth is plotted as a function of log of the solar distance in AU. Circles and dots refer

to observations made before and after perihelion respectively (Jacchia, L.G. 1974. Sky

and Telescope 47, 216; courtesy of Sky and Telescope).

Table 1.1 Parameters for brightness.

Comet mo n

Arend-Rigaux 10.9 4

Tuttle 7.97 6

Kobayashi-Berger-Milon 7.34 3.8West 5.94 2.4

Machholz 6.5 4

Meier 9.3 4Klemola 9.7 4

Meisel, D.D. and Morris, C.S. 1982, InComets, ed Wilkening, Univ. ArizonaPress, Tucson

heliocentric distance r as well as the mean value of n. The value of mo and

n for some comets are given in Table 1.1.

Quite often it is found that it is not possible to find a single value of



n covering the whole range of the Sun-comet distance. In addition, the

variation of the observed brightness, before and after perihelion passage,

seems to require different values of n. Because n varies from comet to

comet, a mean value for n has been derived, with the proper weightage

given to the observations. The results of such a study based on carefully

analyzed photometric data for more than 100 comets grouped into four

classes are shown in Table 1.2. The range of parameters of the classes

selected are,

I (new) 1/a× 106 ≤50(AU)−1; (P ) > 2.8× 106yr

II 5× 10−5 < 1/a < 0.00215(AU)−1; 104 < P < 2.8× 106yr

III 0.00215 < 1/a < 0.01(AU)−1; 103 < P < 104yr

IV 0.01 < 1/a < 0.117(AU)−1; 25 < P < 103yr

Table 1.2 Values of 〈n〉, 〈q〉 and σ(n) for various comets.

Orbit Class I II III IV

Maximum period (yr) ∞ 2.8× 106 104 103

Minimum period (yr) 2.8× 106 104 103 25Pre-Perihelion Dominated

Number of Comets 10 5 5 6

〈q〉(AU) 0.43 0.90 1.17 0.71〈n〉 2.45 3.11 3.32 3.83

σ(n) ±0.35 ±0.71 ±0.52 ±0.49

Post-Perihelion DominatedNumber of Comets 15 16 9 17

〈q〉(AU) 1.20 1.54 0.88 0.80〈n〉 3.16 3.87 4.48 4.94

σ(n) ±0.26 ±0.58 ±0.53 ±0.79

(Whipple, F.L. 1991, In Comets in the Post-Halley Era, Eds. R.L. Newburn, Jr.et al., Kluwer Academic Publishers, p. 1259).

where P is the period. Table 1.2 give weighted mean perihelion distance

〈q〉, the weighted mean 〈n〉 and σ(n) the mean error of 〈n〉. The mean value

of n shows a variation with the comet class and with the age of the comet

(i.e. inversely with the period). This is indicated for both pre-and post -q

data. The post -q observations give a considerably larger value of 〈n〉 than

the pre -q observations. In general, the large value of n required for comets

arises due to the fact that the brightness is the sum total of the reflection

component and the emission of gases from the coma. For the case of pure



reflection the value of n is 2. Therefore the expected brightness of a comet

is a complicated function of the physical condition of the coma. However,

in the absence of any knowledge or data on the comet, one can get an idea

of the average behaviour of its brightness from Eq. (1.3) with a suitable

value of n.

As remarked earlier the larger value of n arises due to the contribution

of the emission component into the total light that is observed. The amount

of emission depends upon the total number of the molecules present in the

coma. In principle, it is possible to predict the brightness of a comet by

relating it to the evaporation of the gases from the nucleus. This is a

difficult practical problem. However, the problem is simplified if the comet

has already been observed once before, for which the visual light curve is

usually available. For such cases, one can use a simplified approach for

predicting the expected brightness for its next apparition. In the visual

spectral region, the emission is mainly due to the C2 molecule. Therefore,

to a first approximation one can assume the visible light is mainly due to the

fluorescence process of the C2 molecule, which depends upon the production

rate of the C2 molecule. This could in turn be related to the total hydrogen

production rate which is very well studied in various comets. From such

a procedure and using the observed light curve, it is possible to evaluate

the unknown constants occurring in the photometric equation. Knowing

the photometric equation, the calculation of the expected brightness of a

comet as a function of r and ∆ is quite simple. (Sec. 6.1.4).

There is still a simpler way than the above method. Since in Eq. (1.3)

the value of n is uncertain, it can be written in a simplified form as

m = 5 log ∆ +m(r) (1.4)

where

m(r) = m0 + 2.5n log r.

From the observed light curve, the value of m(r) can be calculated from

Eq. (1.4) as a function of the time from the perihelion passage (≡ r).

Figure 1.6 shows the results of such calculations for Comet Crommelin

(1984 IV) based on the last four apparitions. Based on Eq. (1.4) and the

curve of Fig. 1.6, the expected brightness can be predicted.

1.7. Main Characteristics

Some of the main observed features in comets are the following:



Fig. 1.6 A plot of the heliocentric magnitude of the Comet Crommelin as a functionof time (equivalent to perihelion distance) from perihelion for four different appearances

(Festou, M. 1983. International Halley Watch News Letter No. 3, p. 4).

Most of the comets seem to deviate from their predicted orbits. They are

known to arrive into the solar vicinity earlier or later than the predictions

based on Newtonian gravitation. The classic example for which the data

exist for the last two centuries is Comet Encke, which has a period of 3.3

years. It arrives earlier every time by about 212 hours. The splitting of

the nuclei of comets into two or more fragments has been seen in many

comets. The best example is Comet West (1975VI) in which the nuclei

split up into four components in a time span of a few days. This can be

seen from Fig. 1.7. There appears to be some correlation between the

time of fragmentation and the increase in brightness of the comet. This

might mean that a spurt of cometary activity leads to fragmentation. For

sun-grazing or planet grazing comets the splitting might also take place

because of the tidal forces. The classic example is the nucleus of Comet

Shoemaker-Levy 9 (D/1993 F2) which is believed to have fragmented into

several pieces primarily due to the tidal forces exerted by Jupiter during

its closest approach in July 1992. The spectacular feature of such an event



Fig. 1.7 Photograph shows the splitting of the nucleus of the Comet West 1975 VI into

four parts in a time period of 10 days. The separation of the various components withtime can easily be seen (Whipple, F.L. 1978. In Cosmic Dust, ed. J.A.M. McDonnell,

New York: John Wiley and Sons, p. 1).

is that as many as 21 nuclei all in a line, were discovered in mid July 1993.

(Fig. 1.8). The subsequent orbit of these fragments and their collision with

Jupiter in July 1994 is schematically shown in Fig. 1.9.

Comets which come very close to the Sun can be completely destroyed.

An example of this type is shown in Fig. 1.10 which was discovered acciden-

tally in the satellite observation and shows the time sequence photograph

of a comet. After its closest approach to the Sun, the comet was not visible

at all, most probably due to complete evaporation or falling into Sun.

Many comets show a sudden increase in brightness of one to two magni-

tudes in a short time scale usually called outbursts or flares. These outbursts

are not periodic in character. An outburst may mean a sudden release of



Fig. 1.8 Image of Shoemaker-Levy 9 taken with the Hubble space Telescope on July 1,

1993. The comets heliocentric distance was 5.46 AU and the geocentric distance was 5.4

AU. The various fragments of the comet can clearly be seen. (Courtesy of H.A. Weaverand collaborators).

material from the nucleus. A classic example in which the flares have been

seen very frequently is Comet Schwassmann-Wachmann 1. Figure 1.11

shows one such event for this comet. This comet has a period of 16.5 years

and has been seen to brighten as much as 8 magnitudes. The flaring activ-

ity appears to be a general property of comets and is not associated with

any particular type of comet. A strong outbursts was seen in Comet Halley

even at a heliocentric distance of around 14 AU.

In many comets successive halos coming out of the coma have also been

seen. Comet Donati (1858VI) is an example of this class where the suc-

cessive halos can distinctly be seen (Fig. 1.12). In many comets a broad

fan-shaped coma coming out of the central condensation has also been seen.

The satellite observations in the Lyman α line of hydrogen at 1216 A

led to the discovery of the enormous extent of the hydrogen envelope (∼



Fig. 1.9 The orbit of the Comet Shoemaker-Levy 9 around Jupiter is shown schemati-cally. The comet which appears to have come close to Jupiter around July 8, 1992 was

disintegrated due to Jupiter’s tidal force. This comet discovered in March 25, 1993 had

as much as 21 components. These components penetrated Jupiter’s atmosphere duringJuly 16 to 22, 1994 (Courtesy of Sekanina, Z., Chodas, P.W. and Yeomans, D.K.).

107 km) around the visible coma of about 104 to 105 km. The size of the

hydrogen halo was found to be larger than the size of the Sun at the same

distance. This can clearly be seen in the observations of Comet Kohoutek

which is shown in Fig. 1.13.

The most characteristic feature of a comet is, of course, the presence

of two tails. One is the dust tail which is curved, also called Type II tail.

The other is the plasma tail which is straight, also called Type I tail or ion



Fig. 1.10 Time sequence photographs taken between August 30, 18h 56mto August 31,

20h 36m by Solwind satellite show the disappearance of the comet. The comet entersfrom the right. The photographs are taken by Howard, R.A., Koomen, M.J. and Michels

D.I. official US Navy photograph. (Courtesy of Howard, R.A. and collaborators).

tail. Figure 1.14 shows these two well-developed tails for the Comet Mrkos

(1957V). Both the tails point to the direction away from the Sun. The

nature of these two tails can be seen clearly in the colour photographs in

which the dust tail appears yellowish and the plasma tail bluish in colour.

Quite often a third short tail has also been seen in the direction towards

the Sun. This is called the anti-tail of the comet. Figure 1.15 shows an

example of this class for Comet Arend-Roland (1957 III).

The dust tail is generally very smooth and structureless. But this is not

so in the case of the plasma tail. Large scale structures of various kinds



Fig. 1.11 An outburst seen in Comet Schwassmann-Wachmann 1. The position of thecomet on October 12, is shown by lines in the margin. The comet is very bright on

October 18, but has become faint by November 3 (Roemer, E. 1966. In Nature and

Origin of Comets. Memoirs of the Society Royale de Sciences of Liege 12, 15; Courtesyof E. Roemer, official US Navy photographs).

have been seen in the plasma tail of comets. As the name indicates, this

tail is composed mainly of ions. Therefore many of the features seen in

the laboratory plasma arising out of various physical processes as well as

others, which cannot be seen in the laboratory plasma, have also been seen

in the plasma tail of comets. For example features like oscillations, kinks,

helices, knots, filaments, rays etc., have been seen in many comets. These

indicate clearly the presence of complex interactions of the magnetic field

with the tail plasma. Some typical large scale observed features in comets

are shown in Figs. 1.16 to 1.18.

1.8. Spacecraft Encounters with Comets

The general nature of comets has been revealed mainly through indirect

means from the studies of ground based observations, rockets, aeroplanes

and satellites. However space mission to comets will provide an opportunity

to make in situ measurements of a comet. Since this is almost like direct

access, it will provide more detailed information about the nucleus. With

this in view several space missions to comets have been undertaken. They

are given in Table 1.3. The main objective of space missions are, to image

the nucleus for its morphology; to determine the composition of gas and

dust; and to understand the interaction of the solar wind with the cometary

ionosphere.



Fig. 1.12 Successive halos around Comet Donati (1858 VI) as observed visually by

Bond on October 1858 (Whipple, F.L. 1981. In Comets and Origin of Life, ed. C.

Ponnamperuma, Dordrecht: D. Reidel Publishing Company, p. 1).

Comet Halley was the first comet to be studied in great detail from

several spacecrafts in 1986. Even prior to Comet Halley observations, in

situ measurements of particles, fields and waves were carried out by the ICE

satellite (International Cometary Explorer). It passed through the plasma

tail of Comet Giacobini-Zinner on September 11, 1985.

During March 1986, around the time of the closest approach of Comet

Halley to the Sun, six spacecrafts from various space agencies made detailed

and extensive in situ measurements of various kinds in the coma of Comet

Halley (Fig. 1.19). All the encounters took place on the Sunward side of



Fig. 1.13 Comet Kohoutek (1973 XII) as seen in visible (left) and in ultraviolet region

taken in Lyman α line (right). Circle represents the apparent size of the Sun at the samedistance. The vast extent of the hydrogen halo can clearly be seen. (Whipple, F.L. 1981,

loc. cit.).

the comet. The closest approach to the nucleus was made by the European

Space Agency’s (ESA) Spacecraft Giotto, which passed at a distance of

approximately 500 km from the nucleus. The Russian spacecrafts Vega

1 and 2 passed at distances of around 8000 km from the nucleus. The

distances of the closest approach of the Japanese spacecrafts Suisei and

Sakigake were around 1.5×105 km and 7.0×106 km respectively. The ICE

spacecraft also passed through at a distance of around 0.2 AU upstream

of Comet Halley later in March 1986. Another space encounter with a

comet took place on July 10, 1992 when the Giotto Spacecraft, which was

successfully redirected in July 1990, passed through the Comet P/Grigg-

Skjellerup nucleus within 100 to 200 km.

The Comet Halley, with a period of about 76 yrs, moves in a retrograde

motion with respect to the planets. Due to the retrograde motion of Comet

Halley with respect to the Earth and the Spacecrafts, the relative encounter

velocity was very high, ∼ 68 km/sec. Since the gas and dust velocities in

the coma are ∼ 1 km/sec, the Giotto Spacecraft essentially saw the static

situation while it passed through the coma as particles hit the spacecraft

from the forward direction. The scientific payload on the spacecrafts, which



Fig. 1.14 Photograph of Comet Mrkos (1957V) taken on 27 August 1957 which shows

the characteristic feature of the two tails, ion tail (straight) and the dust tail (curved).(Arpigny, C. 1977, Proceedings of the Robert A. Welch Foundation Conferences on

Chemical Research XXI, Cosmochemistry, Houston).

passed through Comet Halley, had experiments for the study of flux and

composition of neutrals, ions, electrons and dust, magnetic field and waves,

imaging the nucleus, infrared spectra and ultraviolet images among others.

The space missions, Deep Space 1, Stardust and Deep Impact also

made in situ measurements of various kinds in the Comets 19P/Borrelly,

81P/Wild 2 and 9P/Tempel 1 respectively. The cometary material from

Comet Wild 2 was successfully brought back to the earth by Stardust mis-

sion. This is the first time, laboratory studies are being carried out on the

real cometary dust particles.

The Deep Impact mission released a projectile which impacted on the

surface of the nucleus of Comet Tempel 1 resulting in a crater on the nu-

cleus. As a result, the vapourized ejecta from the deeper layers came out.

This material was subjected to a thorough investigation, in addition to

studying the crater itself. This gave an opportunity for the first time to

study the material in the deeper layers of the nucleus of a comet. All these

missions have met with tremendous success. They provide new insights into

the nature of nucleus, coma, dust and the tails. These observations were

supplemented by extensive observations of various kinds carried out with

worldwide ground based telescopes, satellites, rockets and Kuiper Airborne



Fig. 1.15 Photograph of Comet Arend-Roland (1957III) which shows the anti-tail

(sharp ray towards the left). Taken on 24 April 1957 (Whipple, F.L. 1981, loc. cit.).

Observatory, with sophisticated instrumentation by professionals and am-

ateur astronomers to obtain as much coverage as possible, over the orbit of

the comets. This co-ordinated effort of an unprecedented nature have not

only confirmed our knowledge about comets, theories and hypothesis that

existed before these measurements were made, but have also provided new

and unexpected insight into the cometary phenomena.

1.9. An Overall View

The various aspects of cometary phenomena has come from studies car-

ried out over the entire range of the electromagnetic spectrum from X-rays

to radio region, from in situ measurement of comets and from laboratory

studies of cometary dust particles brought back to earth. All these studies

had a tremendous impact on Cometary Science.

The three major parts of a comet are the nucleus, the coma and the



Fig. 1.16 Comet Kohoutek showing the helical structure. Photograph was taken on 13January 1974. The oscillations in the tail at far off distances from the head can clearly

be seen. (Brandt, J.C. and Chapman, R.C. 1981. Introduction to Comets, Cambridge:

Cambridge University Press, illustrations credited to Joint Observatory for CometaryResearch (JOCR), NASA).

tail (Fig. 1.20). Most of the information about these three components has

come basically from the study of spectra. Almost all the observed activities

seen in a comet should be related directly or indirectly to the nucleus of

a comet. A reasonable working model for the nucleus which is the icy-

conglomerate model was first proposed by Whipple in the 1950’s. In this

model the nucleus was believed to be a discrete rotating body consisting

of frozen water, complex molecules formed out of abundant elements H,

C, N and O, and dust. All the subsequent observations of comets for the

last four decades had supported this model by indirect means, basically

through the observation of the dissociated products of H2O (i.e. OH, H

and O) and H2O+. The first actual detection of H2O in a comet came from

the observation of Comet Halley in 1986, when well resolved rotational

lines of the 2.7 µm band of H2O were detected with observations carried

out with the Kuiper Airborne Observatory. The nucleus contains around

80% of H2O-ice and the rest is made up of other constituents. The single

body nature of the nucleus, in contrast to that of ‘loosely bound system’,



Fig. 1.17 Photograph showing a big knot in the tail of Comet Kohoutek. This is

generally called as ‘Swan-like’ feature. The photograph was taken on 11 January 1974

(Brandt, J.C. and Chapman, R.C. 1981, loc. cit. JOCR photograph).

Table 1.3 Spacecrafts to Comets.

May 12, 2010 15:37 World Scientific Book - 9in x 6in comets


Fig. 1.17 Photograph showing a big knot in the tail of Comet Kohoutek. This is

generally called as ‘Swan-like’ feature. The photograph was taken on 11 January 1974(Brandt, J.C. and Chapman, R.C. 1981, loc. cit. JOCR photograph).

Table 1.3 Spacecrafts to Comets.

Comet Spacecraft Launch Date of Distance of Period of Remarks

date closest closest Cometapproach approach (yr)

(Km)

Halley 6 spacecrafts March 1986 Giotto 76 Out of 6 spacecrafts,

∼500 Giotto mission wasthe closest

Borrelly Deep Space 1 Oct.25, 1998 Sept.22, 2001 ∼2171 6.9 Extended missionto comet after

asteroid flyby

Wild 2 Stardust Feb.7, 1999 Jan.2, 2004 100-150 6.4 Returned to Earth onJan.15, 2006 with

cometary dustTempel 1 Deep Impact Jan.12, 2005 July 4, 2005 ∼500 5.5 104 tons of material

was excavated from

the nucleus

was confirmed by the photographs taken of the nucleus of Comet Halley by

Giotto Spacecraft when it was at a distance of 500 km from the nucleus,

supporting the general concept of the Whipple model. The nucleus was

observed to be irregular in shape and there were only a few active areas on

its surface. The rotation of the nucleus of Comet Halley was also confirmed.

All these results have been confirmed with further observations carried out

with Deep Space 1 on Comet Borrelly; Stardust on Comet Wild 2; and

Deep Impact on Comet Tempel 1. The shapes and sizes of the cometary

was confirmed by the photographs taken of the nucleus of Comet Halley by

Giotto Spacecraft when it was at a distance of 500 km from the nucleus,

supporting the general concept of the Whipple model. The nucleus was

observed to be irregular in shape and there were only a few active areas on

its surface. The rotation of the nucleus of Comet Halley was also confirmed.

All these results have been confirmed with further observations carried out

with Deep Space 1 on Comet Borrelly; Stardust on Comet Wild 2; and

Deep Impact on Comet Tempel 1. The shapes and sizes of the cometary



Fig. 1.18 Photograph of Comet Halley showing well developed streamers. Taken on 8

May 1910. (Report of the Science Working Group, loc.cit.).

nuclei have been determined from the images taken at different angles.

When the comet is far off from the Sun, the continuum spectrum seen

is simply that of the reflected sunlight by the nucleus. As it approaches the

Sun, the gas, mostly made up of complex molecules and the dust are re-

leased from the nucleus, due to solar heating. This then expands outwards

into the vacuum at about 0.5 km/sec giving rise to the observed coma. The

dimension of the visible coma is around 104 to 105 km, while the ultraviolet

coma extends up to about 106 to 107 km. As the gas expands, it is sub-

jected to various physical processes like dissociation, ionization, gas-phase



Fig. 1.19 The geometry of the six spacecraft flybys to Comet Halley. The distances are

marked in logarithmic scale and the Sun is to the left of the Comet. The flyby dates foreach mission are given at the bottom, flyby phase angle in the centre and flyby speeds

at the top (Mendis, D.A. 1988. Ann. Rev. Astron. Astrophys. 26, 11).

reactions, etc. Therefore, the gaseous material in the coma is modified to

a large extent. The complex molecules released by the nucleus, generally

known as parent molecules, ultimately give rise to simple molecules and

radicals like CN, C2, OH, CH, NH2, etc., which are seen in the comet’s

emission spectra. The emission lines of various elements like Na, Si, Ca,

Cr, etc., show up in sun-grazing comets. Through the study of the radio

region, large number of molecules possibly the parent molecules of some of

the observed species, have been identified.

The dust coming out of the nucleus is dragged outwards by the gas ac-

companying it. These dust particles are subjected to the radiation pressure

of the Sun which pushes them in the direction away from the Sun. Since

the dust particles lag behind as they stream away from the sun, they take

a curved path. This gives rise to the observed curved nature of the dust

tail. The tail that is usually seen in the sky is the dust tail, which is made

visible through the scattering of the solar radiation by the dust particles in

the tail. The tail extends up to about 107 km or so.

The understanding of the nature and composition of the dust parti-



Fig. 1.20 Descriptive sketch of a comet.

cles came mostly from the interpretation of continuum, polarization and

infrared radiation from comets. That the grain material could be some

form of silicate, was inferred from the detection of broad emission features

occurring around 10 and 20 µm in comets, which are characteristic features

of all silicate materials. The silicate material could be of the olivine type

which can be further inferred from the detection of a feature at 11.3 µm

in the high resolution spectral observations of Comet Halley. There is an-

other component of the grains in comets came from the discovery of CHON

particles from the in situ measurements of Comet Halley. As the name

indicates, these particles are mostly made up of H, C, N and O. This was

also supported from the ground-based observations of Comet Halley and

other comets by the detection of a broad emission feature around 3.4 µm

attributed to C-H stretching bond of hydrocarbons. Therefore these studies

indicated two major components for the grains in comets, namely silicate

and some form of carbon.

More recent studies of Comets Tempel 1 and Wild 2 has shown the

highly complex mineralogy of the cometary dust particles. They have shown

the presence of silicates of various types, carbonates, water-ice, amorphous

carbon, sulphides among others. The organics present in the dust particles

are also highly complex. The surprising result was the observation of wide

variation in the chemical composition among comets. The dust particles

also contain material of low and high temperature condensates indicating

that there must have been large scale mixing of material in the solar nebula.

The presence of large numbers of small-sized grains, <∼0.1 µm, which cannot



be detected through observations made in the visible region, was found to

exist in comets in abundance. The grains in the coma can also act as a

source of observed molecules which comprise the gas.

The solar radiation also breaks up the original molecules released by the

nucleus and ionizes them. The ionized gas in the coma is swept outwards

by a stream of charged particles present in the solar wind. These two are

coupled through the interplanetary magnetic field. This gives rise to the

plasma tail in the antisolar direction and extends up to about 107 to 108

km. Therefore, the structure and the dynamics of plasma tail are basically

due to the interaction of the cometary plasma with the solar wind. The the-

oretical modelling of the interaction between the solar wind plasma and the

cometary ions had predicted the existence of several gross features like bow

shock, ionopause etc., which were confirmed based on the in situ measure-

ments carried out on Comets Halley, Giacobini-Zinner, Borrelly, Tempel 1

and Wild 2 by the spacecrafts. Therefore the multi-layered structure and

variations seen in comets arise due to complex interactions taking place in

the plasma between electrons, heavy cometary ions, cometary protons and

solar wind protons as they are being thermalised at different positions in

the coma. The spectrum of the plasma tail shows mainly emissions from

ions like H2O+, CO+, CO+2 , N+

2 , etc; of these the emission due to CO+ is

the dominant one. Since this emission spectrum lies in the blue spectral

region, the ion tail appears blue in colour photographs.

The comets are believed to be members of the solar system inferred

from the observed orbital characteristics of comets. The study based on

the isotopic ratio of 12C/13C in many comets gives a value ∼ 90 which

is the same as the solar system value. Several other isotopic ratios seen

in Comet Halley are also consistent with the solar system values. These

results seem to suggest that the cometary material and the solar system

material are similar in nature. With regard to the origin of comets, the

widely accepted hypothesis is that of Oort. He pointed out that there

appears to be a cometary reservoir whose aphelia is about 50,000 AU or

more from the Sun. This is usually called the Oort cloud. It is estimated

that there may be around 1012 comets in the Oort cloud. Many of the

comets leaving this cloud are finally brought into the solar system due to

stellar perturbations. This accounts for the steady influx of comets into

the solar system. However, the long period comets coming from the Oort

cloud faces serious problems in explaining the number and inclination of

the observed short period comets. This has led to the hypothesis that the

short period comets come from another population of comets from a region



beyond the orbit of Neptune, between 30 and 50 AU. This region is generally

called the Kuiper belt. The origin of comets therefore is related to the origin

of the Oort cloud and the Kuiper belt. Several hypotheses have been put

forward to explain the origin of these clouds.

In the succeeding chapters, we would like to elaborate on some of these

aspects. Before going into the actual subject matter, we would like to give

a brief account of the physical background required for the interpretation

of various observed phenomena.

Problems

1. What is the basis of the assumption that a comet possesses a nucleus

at its centre? Since it is hard to see the nucleus of a comet directly,

what is the best way to locate it?

2. Suppose a Comet A has n = 3 [Eq. (1.2)] in its 50th orbit around the

Sun while Comet B has n = 6 in its first orbit around the Sun. Does

one expect the same brightness variation of Comets A and B in their

next orbits? Give reasons. Explain why the two values of n could be

vastly different.

3. Is it possible for the comets to be in orbit around the Sun, but not seen

from the Earth?

4. Discuss with examples that impact of comets on solar system objects

is a natural phenomena.

5. Discuss the consequences of a comet hitting the Earth.

6. Compare the energy released by a 1 km size comet moving at 60 km/sec

and suddenly coming to a stop with that of two 3500 lb cars colliding

head-on at 50 km/hr.

References

Some of the references pertaining to cometary studies are the following:

1. Bailey, M.E., Clube, S.V.M. and Napier, W.M. 1990. The origin of

comets. Pergamon Press, Oxford.

2. Brandt, J.C. and Chapman, R.D. 2004. Introduction to comets. 2nd

Edition, Cambridge University Press, Cambridge.

3. Festou, M.C., Keller, H.U. and Weaver, H.A. (eds), 2005. Comets II,

Univ. Arizona Press, Tucson.



4. Grewing, M., Praderie, F. and Reinhard, R. (eds). 1987. Exploration

of Halleys Comet. Springer-Verlag, Berlin. (Astron. Astrophys., 187,

1-936, 1987).

5. Huebner, W.F. (ed.) 1990. Physics and Chemistry of Comets.

Springer-Verlag, Berlin.

6. Lazzaro, D., Mello, S.F. and Fernandez, J.A. (eds.) 2006. Asteroids,

Comets and Meteors, Cambridge University Press, Cambridge.

7. Nature. 1986. Encounters with Halley, 321, 259.

8. Newburn, Jr., R.L., Neugebauer, M. and Rahe, J. (eds.). 1991. Comets

in the Post-Halley Era. Vols. 1 and 2. Kluwer Publishers, Dordrecht.

9. Wickramasinghe, N.C. 2009. Astrobiology, Comets and the Origin of

Life, World Scientific Publishing Co., Singapore.

10. Wilkening. L.(ed.). 1982. Comets, Univ. Arizona Press, Tucson.

Deep Impact: The important and first of its kind results based on Deep

Impact studies of Comet Tempel 1 are presented in the following publica-

tions:

11. Deep Impact 2005. Science (Special Issue), 310, 257-283.

12. A’Hearn, M.F. and Combi, M.R. (Eds). 2007. Deep Impact Mission to

Comet 9P/Tempel 1, Icarus (Special Issue), 187, 1-356 (Part I); 190,

283-459 (Part II).

Stardust: The important and first of its kind results based on in situ mea-

surements and laboratory studies of collected dust particles from Comet

Wild 2 from Stardust mission are presented in the following publications:

13. Stardust 2004. Science (Special Issue), 304, 1762-1780.

14. Stardust 2006. Science (Special Issue), 314, 1707-1739.


CHAPTER 2

Dynamics

2.1. Orbital Elements

The objects in space are generally specified with respect to the ecliptic

or to the equatorial system of coordinates. In the former the Earth’s orbit

around the Sun, i.e., the ecliptic plane is the reference frame, while in the

latter it is the plane of the Earth’s equator. The position of an object is

specified by the longitude and the latitude in the ecliptic system and by the

right ascension (α) and the declination (δ) in the equatorial system. The

right ascension is measured from the vernal equinox, which is the point

where the ecliptic plane cuts the Earth’s equator. The declination is the

angular distance from the north to the south of the celestial equator. The

two systems can be transformed from one to the other through trigonomet-

ric relations.

The orbit of a body around the Sun is generally a conic section. In

general, the conics are the ellipse, parabola and hyperbola. The major axis

refers to the maximum diameter of the ellipse and it determines the size of

the ellipse (Fig. 2.1). The eccentricity e of the ellipse is defined as the ratio

of the distance between the center and a focus to the length of the semi

major axis. An ellipse is completely defined by the eccentricity and the

major axis. The value of e varies between 0 and 1. A circle is a special case

of the ellipse when the eccentricity is zero. The parabola has an eccentricity

equal to unity while a hyperbola has an eccentricity greater than one. In

an elliptical orbit the closest and the farthest distance of the object from

the Sun which is stationed at one of the foci is known as the perihelion and

aphelion distances respectively.

The position of a comet in the sky is generally referred to the ecliptic

system of coordinates. To define completely an orbit in space, six quantities

33



Fig. 2.1 Various parameters defined for an elliptical orbit. S denotes the position ofthe Sun occupying one of the foci.

Fig. 2.2 The orbital elements required for specifying an orbit.

usually termed as elements of the orbit are to be specified (Fig. 2.2). They

are the following:

a = length of the semi major axis;

e = eccentricity;

i = angle between the orbital plane and the plane of the ecliptic;

Ω = longitude of the ascending node. This is the angle from the vernal


Dynamics 35

equinox along the ecliptic plane to the point of intersection of orbital

plane with the ecliptic plane;

ω = argument of the perihelion, which is the angle measured from the

ascending node to the perihelion point.

The first two parameters specify the size and the shape of the orbit in

its orbital plane, while the other three define the orientation of the orbit

with respect to the ecliptic plane.

The sixth element is the time parameter which defines the position of the

body in its orbit at that time. This is taken to be T , the time of perihelion

passage. This gives a reference time to fix the body at other times in its

orbit.

Therefore, the quantities, a, e, T and the angle i, ω,Ω define completely

the position of the body and its orbit at any given time. For a parabolic

orbit, the semi major axis which is infinite is replaced by the perihelion

distance q, which defines the size of the parabola.

2.2. Orbit in Space

2.2.1. Relevant equations

The orbit of a comet is a conic section about the Sun and it can be

defined under the Newtonian Gravitation.

The equation of motion of a comet of mass m around the Sun can be

represented as (d2r

dt2

)= −

(G(M +m)

r3

)r ≡ µr

r3(2.1)

where M is the mass of the Sun and G is the gravitational constant. Since

m <∼M, µ = GM and r is the position vector of the comet relative to

the Sun, the use of plane polar coordinates r and θ in the orbital plane

of motion allows one to separate the Eq. (2.1) into r and θ components,

giving, for the r component

r − rθ2 = − µr2

(2.2)

and for the θ component

rθ + 2rθ = 0 (2.3)

Equation (2.3) can be written as

1

r

d

dt(r2θ) = 0. (2.4)



The integration of the above equation gives the specific (means per unit

mass) angular momentum integral

r2θ = constant = h. (2.5)

Specific energy integral is given by

E′ =1

2(r2 + r2θ2)− µ

r. (2.5a)

From the above system of equations, one can derive the polar equation for

the conic section for an angle θ as

r =p

1 + e cos(θ − ω)(2.6)

where ω is constant, p = h2/µ is the semi latus rectum and e =√

1 + 2E′h2

µ2

is the eccentricity of the orbit and therefore the length of the semi major

axis a = − µ2E′ . ω is actually the angle that the major axis makes with the

axis θ = 0. At the perihelion point θ = ω = 0 and therefore the perihelion

distance is given by h2/µ(1 + e). The perihelion distance is also equal to

a(1− e). Hence

h2 = µa(1− e2). (2.6a)

Equation (2.6) can be expressed as

r =a(1− e2)

1 + e cos(θ − ω)=

a(1− e2)

1 + e cos f(2.7)

where a is the semimajor axis and the angle f is known as the true anomaly

(Fig. 2.3).

The other quantity of interest is the velocity of the object in its orbit.

From Eqs. (2.1) and (2.4), the expression for the orbital speed of the particle

can be obtained and it is given by

v2 = µ

[2

r− 1

a

]. (2.8)

The quantity 1/a is +ve , zero or −ve depending on whether the orbit is

an ellipse, a parabola or a hyperbola. The velocity at the perihelion point

for an ellipse is given by

v2p =

µ

a

(1 + e

1− e

); (2.9)

similarly at the aphelion point, the expression for the velocity is given by

v2a =

µ

a

(1− e1 + e

). (2.10)


Dynamics 37

Fig. 2.3 Definition of eccentric anomaly E. f represents the true anomaly.

From Eq. (2.7) it is clear that the value of 1r monotonically decreases from

perihelion to aphelion, Eq. (2.8) would then suggest that the velocity of

the object is maximum at perihelion and minimum at aphelion and varies

along its orbit in between these, two limits.

Another quantity of interest is the heliocentric radial velocity of the

object. For elliptical orbits, it can be shown from Eq. (2.7) by taking its

time derivative that

r = ∓µ1/2

[a2e2 − (a− r)2

ar2

]1/2

. (2.11)

The minus and positive signs refer to preperihelion and postperihelion radial

velocities.

The mean angular motion n of the body in its orbit with the period P

is by definition

n =2π

P. (2.12)

Since Eq. (2.5) suggest that h = r2θ = constant and 12r

2θ is the areal

velocity, the orbital period

P =Area of the ellipse

areal velocity

=2πa2

√1− e2

h(2.12a)



P = 2π

√a3

µ. (2.13)

Hence

n = µ1/2a−3/2. (2.14)

If T represents the time of perihelion passage, the angle swept by the

radius vector in a time interval (t− T ) is called the mean anomaly M and

is defined as

M = n(t− T ). (2.15)

If a circle is drawn with OB as radius, then the line FP referring to the

ellipse when extended perpendicular to the major axis cuts the circle at

T ′ (Fig. 2.3). The angle T ′ OB is called by definition eccentric anomaly,

generally denoted as E. Since PF/T′F=√

1− e2 for any point P on the

orbit, it can be shown that

SF = x = a cosE − ae

PF = y = a√

1− e2 sinE

Therefore

r =√x2 + y2 = a(1− e cosE) (2.16)

. Further

cos f = 1− 2 sin2 f

2(2.16a)

or

2r sin2 f

2= r(1− cos f) (2.16b)

However

cos f =SF

r=

a cosE − aea(1− e cosE)

(2.16c)

and

sin f =PF

r=a(1− e2)1/2 sinE

a(1− e cosE)(2.16d)

Therefore the Eq. (2.16b) can be written as

2r sin2 f/2 = a(1 + e)(1− cosE) (2.16e)


Dynamics 39

similarly

2r cos2 f/2 = a(1− e)(1 + cosE)

Therefore

tan f/2 =

(1 + e

1− e

)1/2

tanE

2(2.17)

This is the relation which connects the eccentric anomaly E with the true

anomaly f .

There is also a relation connecting the mean anomaly M and the eccen-

tric anomaly E. This is generally referred to as Kepler’s equation which is

derived as follows:

The quantity (t−T )/P represents the fractional area of the ellipse swept

by SP with respect to the point B, where P is the orbital period. This is

also equal to n(t− T )/2π. From the law of areas and the properties of the

auxiliary circle, it follows that

n(t− T )

2π=M

2π=

areaBSP

area of ellipse=

areaBST ′

area of circle

But AreaBST ′ = area of circular sector BOT′- area SOT′

=a2E

2− a

2ae sinE

Therefore

M

2π=a2

2

(E − e sinE)

πa2

or

E − e sinE = M (2.18)

which is the Kepler’s equation.

The above relation can also be written as

E − e sinE = n(t− T ). (2.19)

The corresponding relations for the parabolic case are

r =2q

1 + cos f(2.20)

v2 =2µ

r(2.21)

(µ

2q3

)1/2

(t− T ) = tanf

2+

1

3tan3 f

2(2.22)



where q = a(1− e), and for the hyperbolic case,

r =a(e2 − 1)

1 + e cos f(2.23)

r = a(e coshF − 1) (2.24)

tanhF

2=

(e− 1

e+ 1

)1/2

tanf

2(2.25)

v2 = µ

(2

r+

1

a

)(2.26)

e sinhF − F = M =( µa3

)1/2

(t− T ). (2.27)

2.2.2. Orbital elements from position and velocity

The position and the velocity of an object along its orbit can be obtained

from the solution of the above equations. The reverse problem which is

often of interest in cometary studies is to determine the elements of the

orbit from a given set of position, velocity and time.

Let the position of the body in the heliocentric coordinate system at a

time t be (x, y, z) and the velocity components (x, y, z). Then

r2 = x2 + y2 + z2 (2.28)

and

v2 = x2 + y2 + z2. (2.29)

Since r and v are known, the semimajor axis can be calculated from

Eq. (2.8).

The areal constant h may be regarded as the vector product of r and

the orbital velocity v of the comet relative to the Sun. If the components

of h along x, y and z directions are represented as hx, hy and hz then,

hx = yz − zyhy = zx− xz (2.30)

hz = xy − yx

and

h2 = h2x + h2

y + h2z = µp.


Dynamics 41

Therefore the value of p the length of the semilatus rectum can be calcu-

lated. From a knowledge of p and a, e can be obtained from the relation

p = a(1− e2). (2.31)

The projection of h onto the three planes yz, zx and xy gives the following

expressions

h sin i sin Ω = ±hxh sin i cos Ω = ∓hy (2.32)

h cos i = hz.

Hence, the above equations give i and Ω. The upper sign and lower sign

refer to the cases when i is less than or greater than 90.

The value of (ω+f) can be derived from the following relations whichre-

late the position of the point (x, y, z) in terms of the angles of the orbit.

sin(ω + f) =z

rcosec i (2.33)

and

cos(ω + f) =1

r(x cos Ω + y sin Ω).

The value of f can be obtained from the relation

r =(h2/µ)

1 + e cos f. (2.34)

Hence ω can be determined. The only other quantity remaining to be

determined is the time of the perihelion passage T , which depends upon

the conic section. For an elliptical orbit the eccentric anomaly E can be

obtained from the Eqs. (2.16) or (2.17). knowing E, e, n and f , the value

of T can be calculated from the Eq. (2.19).

Therefore all the elements a, e, i, ω,Ω and T can be determined. The

procedure for the other two types of orbits are also similar. For parabolic

and hyperbolic orbits, the Eqs. (2.22) and (2.27) may be used to evaluate

T .

2.2.3. Orbital elements from observations

Since in general six elements are required to specify completely the orbit

in space, it follows that six independent quantities must be obtained by the

observations. A single observation gives only two quantities say, in terms

of the angular coordinates α and δ of the body. Therefore in all, three

different sets of observations are required to define its orbit.



Let the heliocentric equatorial rectangular coordinates of the comet and

the Earth at any given time be denoted by (x, y, z) and (X,Y, Z) respec-

tively with respect to the plane of the celestial equator. Let their heliocen-

tric distances be r and R which are given by

r2 = x2 + y2 + z2

and

R2 = X2 + Y 2 + Z2. (2.35)

Then neglecting comet’s mass,

x = −GMx

r3(2.36)

and for the Earth

X = −G(M +me)X

R3(2.37)

where M and me are the mass of the Sun and the Earth respectively.

Let the geocentric direction cosines of the comet be l,m and n and its

geocentric distance be ρ. Then

x = X + lρ

y = Y +mρ

and

z = Z + nρ. (2.38)

The direction cosines l,m and n are given in terms of the observed position

of the comet, right ascension α and declination δ, as

l = cosα cos δ

m = cos δ sinα

and

n = sin δ. (2.39)

Equations (2.36), (2.37) and (2.38) give

ρl + 2ρl + ρl = −GM (X + lρ)

r3+G(M +me)

X

R3(2.40)

or (ρ+

GMρ

r3

)l + 2ρl + ρl = −GX

(M

r3− M +me

R3

). (2.41)


Dynamics 43

Fig. 2.4 Shows the geometry of the triangle with distances marked.

There will be two other equations in Y and Z. These equations may be

solved to give (ρ+ (GMρ/r3)), 2ρ and ρ. Except for r, all other quantities

are known or can be derived from the observed quantities; r can also be

expressed in terms of ρ. From the triangle whose sides are R , r and ρ

(Fig. 2.4) the following relation may be obtained:

r2 = R2 + ρ2 − 2ρR cos θ (2.42)

where θ is the angle between R and ρ. The projection of R in the direction

of ρ gives

R cos θ = −(lX +mY + nZ). (2.43)

Therefore

r2 = R2 + ρ2 + a1ρ (2.44)

where

a1 = 2(lX +mY + nZ).

Equations (2.41) and (2.44) can be solved for r and ρ. Knowing the values

of r and ρ, the comet’s heliocentric coordinates (x, y, z) and the velocity

components (x, y, z) can then be obtained from the relations

x = X + lρx = X + lρ+ lρ (2.45)

and with similar equations for y, y, z and z. Knowing the position and

velocity of the comet, the orbital elements and hence the orbit can be



determined from the method already discussed. The method outlined above

is generally referred to as Laplace’s method.

The actual computation involves a knowledge of the value l,m, n,X, Y, Z

and their derivatives. The values of X,Y, Z are given in the Ephemeris.

From this data the values of the first derivative can be found out. The

calculation of geocentric direction cosines, their first and second derivatives

can be deduced from three observations of the comet which are not too far

off from each other. Let the dates of observation be t1, t2 and t3. The right

ascension and declination for these three times are known. Here we will just

show an approximate method of getting the first and second derivatives. In

actual practice one can use various refined methods.

The average value of the first derivative of l for the time between t1 and

t2 is given by

l12 =l2 − l1t2 − t1

. (2.46)

Similarly

l23 =l3 − l2t3 − t2

. (2.47)

If the time interval (t2 − t1) ≈ (t3 − t2), then the value of l at time t2 is

approximately equal to

l2 =1

2

[l12 + l23

]. (2.48)

Similarly

l2 =l23 − l12

12 (t3 − t1)

. (2.49)

Similar relations for the first and second derivatives and m and n can be

obtained.

The elements obtained from three sets of observations define the initial

orbit of the comet. For getting a better orbit it is necessary to have many

more observations. The initial orbit can be improved further as more and

more observations become available. In fact equations can be set up for

the difference between the predicted and the observed positions. These can

then be solved to get the corrections for the preliminary orbit elements.

This procedure will eventually lead to a more accurate orbit. Such an orbit

is called a ‘definitive orbit’. The orbit calculated based on the six elements

gives the position of the comet in space. In order to find the position of the

comet in the plane of the sky, it is necessary to know the position of earth


Dynamics 45

in space. The geometrical position of the earth can be obtained from the

‘ephemeris’ which is used to find the position of the comet in the plane of

the sky.

In the discussion so far, it is assumed that the comet is only under the

influence of the Sun’s gravitational field. But in actual practice the orbit

gets perturbed due to the planets as comets enter the solar system. The

dominant effect arises mainly from the planets Jupiter and Saturn because

of their large masses. When the comet is far off, the perturbation produced

due to stars, has also to be considered. The calculations which include

many of these perturbations have been carried out numerically. Through

these efforts the dynamical evolution of comets has been studied.

Problems

1. The components of velocity of a body are (0, 1, 3) corresponding to the

position (4, 2, 1). Calculate a and e. What is the nature of the orbit?

Assume for simplicity µ = 1.

2. The ecliptic heliocentric coordinates of position and velocity of a comet

are (4, 2, 3) and (2, 2, 1) respectively on March 16, 1959. Find the

elements of the orbit of the comet. Here again assume µ = 1.

3. The time period of the Earth around the Sun is 1 year and its orbital

velocity is 30 km/sec. Compute the distance from the Earth to the

Sun.

4. Calculate the lifetime of comets which have aphelion distances of 5 AU

and 5 × 104 AU, are almost in parabolic orbits and can survive 1000

perihelion passages.

5. The comet moving in an elliptical orbit has an eccentricity of 0.985.

Compare its velocity at perihelion and aphelion.

6. Show that in elliptic motion about a focus under attraction µ/r2, the

radial velocity is given by the equation

r2r2 =µ

aa(1 + e)− r r − a(1− e)

7. Calculate roughly the distance from the Sun beyond which Comet Hal-

ley spends about half of its total time period of 76 yrs.

8. Estimate the average values of r, dr/dt, dv/dt and the kinetic energy

in an elliptical orbit taking time as an independent variable.

9. A satellite is orbiting in a circle at an altitude of 600 km. Knowing the

radius and surface gravity of the Earth, calculate its orbital velocity



and period of revolution. If it is brought to an altitude of 400 km,

what is the time period?

References

The solutions of the equations of Sec. 2.2.3 are specially discussed in:

1. Moulton, F.R. 1970. An Introduction to Celestial Mechanics, New

York: Dover Publications Inc.

2. Roy. A.E. 1965. The Foundations of Astrodynamics, New York: The

Macmillan Company.

3. Danby, J.M.A. 1988. Foundations of Celestial Mechanics, Richmond:

Willmann-Bell, Inc.

Orbit calculation with personal computer

4. Boulet, D.L. 1991. Methods of Orbit determination with Micro Com-

puter: Richmond: Willmann-Bell, Inc.


CHAPTER 3

Physical Aspects

The gaseous material of a comet is immersed in the radiation field of the

Sun. The study of the interaction between the two requires knowledge of

some of the basic laws of radiation as well as of the spectroscopy of atoms

and molecules. Here we will briefly review some of the relations which are

used in subsequent chapters for interpreting the cometary observations.

3.1. Black Body Radiation

The energy radiated at different wavelengths by a black body (which

absorbs all of the incident radiation) at temperature T is given by Planck’s

law

Bλdλ =2hc2

λ5

1

ehc/λkT − 1dλ (3.1)

where h, k and c denote the Planck constant, Boltzmann constant and the

velocity of light respectively. Figure 3.1 shows a plot of the Eq. (3.1) for

several temperatures which shows clearly the shift of the maximum of the

curve to shorter wavelengths with an increase in temperature. The wave-

length corresponding to the peak of the curve λmax can be represented by

the equation

λmaxT = 0.2897 (3.2)

known as Wien’s displacement law. Figure 3.1 also shows that the total

amount of the radiation emitted shifts gradually from the ultraviolet to the

visible and to the infrared spectral regions as the temperature goes from a

higher value to a lower value.

If the photon energy hν is very much less than the thermal energy i.e.

hν kT , the Planck’s formula simplifies to

47



Fig. 3.1 The Planck function for various temperatures. The regions of ultraviolet,visible and infrared are clearly marked.

Bν(T ) =2ν2kT

c2(3.2a)

This is the well known Rayleigh - Jeans law which is valid for long wave-

length region, such as radio wavelength region. This is extensively used in

the study molecules in the radio region.

The energy density uν of the radiation field is related to Bν through the

relation

uν =4π

cBν(T ) (3.3)

The total energy density is given by


Physical Aspects 49

u =

∫uνdv = aT 4 (3.4)

where a is the radiation constant. The relation (3.4) known as the Stephan-

Boltmann law, shows that the energy density of the black body radiation

depends upon the fourth power of temperature.

3.2. Perfect Gas Law

The particles of gas in a container which are constantly in motion collide

with each other as well as with the walls of the container, giving rise to

a resultant force and hence the pressure. The pressure is a function of

the density of the gas and the temperature. For a perfect gas, in which

the interatomic or intermolecular forces can be ignored, there is a simple

mathematical relation between the pressure, density and temperature of

the gas, called the equation of state. The equation of state for a perfect gas

is given by

p = nkT (3.5)

where p, n and T represent the pressure, density and temperature, respec-

tively.

For a gas containing a number of non-interacting species of various types

each exerting its own pressure, the total gas pressure is just the sum of the

various components, i.e.

p =∑

pi =∑

nikT. (3.6)

3.3. Dissociative Equilibrium

If the gaseous medium is at a low temperature, the atoms can combine

together to form molecules. These molecules in turn may dissociate giving

back the atoms. Finally an equilibrium situation will be reached when the

direct and the reverse processes balance each other. For such an equilibrium

situation, it is possible to calculate the number of molecules formed out of

the individual atoms for a given temperature.

Consider two atoms x and y combining together to form the diatomic

molecule xy, i.e.

x+ y xy.



For the equilibrium situation one has the relation

p(x)p(y)

p(xy)= K(xy) (3.7)

where p(x), p(y) and p(xy) are partial pressures of x, y and xy respectively.

K(xy) is called the equilibrium constant or the dissociation constant of the

reaction. The equilibrium constant depends upon the temperature and on

various parameters of the molecule. An expression for the equilibrium con-

stant can be obtained in an explicit form as

log10Kxy(T ) = log10

pxpypxy

= −5040.4D

T+

5

2log10T

+3

2log10M + log10

QxQyQxy

+ 4.41405. (3.8)

HereM is the reduced mass of the molecule equal to (mxy/(mx+my)) where

mx,my and mxy are the masses of x, y and xy respectively. The Q’s are the

partition functions and D is the energy required to dissociate the molecule

called the dissociation energy. For many molecules of astrophysical interest

the equilibrium constant can be calculated from Eq. (3.8) as all the relevant

spectroscopic paramters are known. Since the equilibrium constant is a

function of the temperature, one usually fits the calculated data with a

polynomial expression of a suitable form.

3.4. Doppler Shift

The frequency of the emitted radiation depends upon the relative ve-

locity of the source and the observer. The effect is produced only by the

component of velocity in the direction towards or away from the observer

called the radial velocity. The shift of the lines produced as a result of the

above motion is termed as Doppler shift or Doppler effect. The expected

shift for a source moving with the velocity v is given by

∆λ

λo=

(λ− λo)λo

=v

c(3.9)

where λ and λo are the observed and the laboratory wavelength respectively.

The relative velocity of the source is denoted as positive if it is moving away

from the observer and negative if it is moving towards the observer. In the


Physical Aspects 51

former case, the shift is towards longer wavelengths while in the latter case

it is towards shorter wavelengths. From the measurement of the shift of the

lines using Eq. (3.9), it is possible to determine the velocity of the source.

Doppler shift has been used extensively in astronomy to derive the relative

velocities of the various astronomical objects.

3.5. Spectroscopy

3.5.1. Atomic spectroscopy

Bohr’s formalism of the absorption and emission processes in atoms is

that the lines arise out of transitions between well-defined electron energy

levels having definite quantum number. The wavelength of the emitted

radiation arising out of the two energy levels E1 and E2 (E2 > E1) is rep-

resented by

λ =hc

E2 − E1(3.10)

When the transition takes place from a lower level to a higher level, the

energy is lost from the incident radiation and it gives rise to an absorption

line. If the reverse process takes place it releases the energy and is kown as

the emission line.

The energy level diagram for the hydrogen atom, which has a single

orbital electron around a proton, is shown in Fig. 3.2. The energy levels

are defined by the principle quantum number n which can take values n =

1, 2, · · · ,∞. The Lyman series arises out of the transition from n = 2, 3, · · ·to n = 1. The Lyman α line corresponding to transition n = 2 to n = 1 has

a wavelength of 1216 A and is in the ultraviolet region. This line is very

strong in comets. Most of the lines of the Balmer series lie in the visible

spectral region. Similarly there are Paschen, Brackett and other series,

whose lines lie mostly in the infrared region.

As n increases, the energy levels come closer and closer together and

finally they coalesce. The transitions arising out of these highest levels

give rise to a continuum. The excitation potential of a line is the energy

required to excite the line. The excitation potential for the Lyman α line

is 10.15 eV. The ionisation potential is the energy required to remove the

electron completely from the atom. For the hydrogen atom this energy is

13.54 eV. For atoms with more electrons, the spectra becomes complicated.



Fig. 3.2 Energy level diagram of the hydrogen atom showing various transitions.

However, the basic model on which they can be explained remains the same

except that one has to consider various types of transitions.

Since the electrons in an atom have orbital and spin angular momenta

around the nucleus, they are therefore characterised by three quantum num-

bers, n, l and j. They denote total quantum number, the orbital angular

momentum quantum number and the total angular momentum quantum

number. The quantum number j for the case of l and s coupling is given

by

j = l + s


Physical Aspects 53

where s is the spin quantum number representing the spin of the electron

and can take the values +12 and − 1

2 . For many electron systems, the vec-

tor sum of the above quantities has to be taken and is represented by the

capital letters L, S and J . Therefore

L =∑

li, S =∑

si, and J = L+ S.

The J levels are in general degenerate with (2J + 1) levels. They split up

into (2J + 1) levels in the presence of an external magnetic field. L can

take values 0, 1, 2 up to (n − 1) and they are represented as S, P, D, F,...

terms. The value with S = 0, 12 , 1, · · · denotes the multiplicity of the levels

and refers to lines as singlets, doublets, triplets, etc. The level is generally

written as

n(2S+1)LJ

where n is the total quantum number, (2S+1) gives the multiplicity, J the

total angular momentum and L is the term symbol.

In general, the transitions between various levels have to satisfy certain

selection rules. For electric dipole transitions the selection rules are the

following:

∆J = 0,±1 with J = 0 9 J = 0

∆L = 0,±1

∆S = 0.

In situations where dipole transitions are forbidden it is possible to

observe magnetic or quadrupole transitions. These are called forbidden

lines. The transition probabilities of these lines are much smaller than those

of allowed transitions. Many of the forbidden lines have been observed in

various astrophysical situations because of low density present in them.

Collisions are infrequent in such an environment and hence forbidden lines

can be seen. Many of these lines cannot usually be observed under normal

laboratory conditions. The forbidden lines are generally denoted with a

square bracket. For example the forbidden line λ = 6300 A of oxygen

arising out of 1D level is denoted as O[1D]. If the atom is neutral, it is

designated by putting I in front of the chemical symbol, like OI, NI, ... The

symbols II, III,... represent atoms in singly ionized, doubly ionized states

like NII, NIII,...



3.5.2. Molecular spectroscopy

Many of the diatomic and complex molecules are abundant in objects

like the Sun, the cool stars and comets. If the two atoms in the diatomic

molecule are of the same type, it is called a homonuclear molecule. Exam-

ples of this type are H2, N2, O2 etc. If the two atoms are of different types,

like CN, CH, OH, etc., it is called a heteronuclear molecule. The spectrum

of even the simplest diatomic molecule shows complicated behaviour com-

prising different bands and each band itself is made up of many lines. This

is due to the fact that the two atoms in the molecule can vibrate individu-

ally along the common axis as well as rotate along the axis perpendicular

to the common axis. The total energy of the molecule is the sum total of

the kinetic and potential energies of the electrons and the nuclei. This is

generally represented in terms of the potential energy curves which give the

variation of the potential energy as a function of the internuclear separa-

tion. The energy required to separate a stable molecule into its components

is called the dissociation energy of the molecule. The various vibrational

energies of the molecule are denoted by the vibrational quantum number

v and can take values 0, 1, 2,... Each vibrational level is further split up

into various rotational levels denoted by the rotational quantum number J .

Therefore each electronic state has many vibrational levels, each of which

in turn has several rotational levels as shown schematically in Fig. 3.3. A

transition can take place between vibrational and rotational levels of the

two electronic states. The total energy E, of the molecule can be repre-

sented as a sum of the electronic (Ee1), vibrational (Evib) and rotational

(Erot) energies, namely

Etotal = Ee1 + Evib + Erot. (3.11)

In terms of actual energies

Ee1 > Evib > Erot.

The electronic transitions give lines which fall in the visible and UV regions

while those of rotational transitions lie in the infrared and far-infrared re-

gions. The vibrational energy of the molecule can be represented as

G(v)hc ≡ Evib = hcwe

(v +

1

2

)− hcwexe

(v +

1

2

)2

+hcweye

(v +

1

2

)3

+ · · · (3.12)


Physical Aspects 55

Fig. 3.3 Schematic representation of the vibrational and rotational levels of two elec-

tronic states A and B of a molecule. The left one shows the transitions involving P, Qand R branches. The right one shows the transitions which define the band sequence

(see text).

where we, wexe and weye are the spectroscopic constants of the molecule.

The expression for the rotational energy is of the form

F (J)hc = Erot =h2

8π2IJ(J + 1) = hcBJ(J + 1) (3.13)

where B = [h/(8π2cI)], J is the rotational quantum number and it can take

values, 0, 1, 2, · · · I is the moment of inertia of the molecule. B is known as

the rotational constant of the molecule. All the quantum numbers of the

lower electronic level (T ′′) are denoted as double prime (′′) , like v′′ and J ′′,

while that of the upper level (T ′) as prime (′) like v′ and J ′. Therefore, the

wave number (cm−1) of the transition between the two electronic states is

given by

ν =1

hc[Eel(T

′)− Eel(T′′)] + [Evib(v′)− Evib(v′′)]

+[Erot(J′)− Erot(J

′′)]. (3.14)



The classification of the electronic states of a molecule is based on a

scheme similar to that employed for atoms. L, i.e., (L1 + L2) and S, i.e.,

(S1 +S2) are the total angular momenta of the two atoms in the molecule.

The projections along the axis of the molecule are denoted as Λ and Σ. Λ

can take values 0, 1, 2, · · · , L. The designations used for these states are

for Λ = 0 1 2 3 · · ·Σ Π ∆ Φ · · · (for molecules)

similar to S P D F · · · in the atomic case.

The total angular momentum Ω is given by

Ω = Λ + Σ

which is similar to the case of quantum number J in the atomic case. The

selection rules for the electronic transitions are

∆Λ = 0,±1

∆Σ = 0

and

∆Ω = 0,±1.

The transition between any two electronic states, is determined by the

vibration-rotation structure of the two states involved. There are no rigor-

ous selection rules for the vibrational quantum number and so the transi-

tion can take place between any two vibrational levels of the two electronic

states. A pure vibrational transition between the two electronic states,

called a band is denoted as (v′, v′′), i.e., the quantum number of the upper

level is written first. Thus for example (2, 0) means a transition from the

upper vibrational level v′ = 2 to the lower vibrational level v′′ = 0. Pure vi-

brational transitions in a given electronic state are allowed for heteronuclear

molecules but not for homonuclear molecules like C2, N2, etc.

Each of the vibrational bands is further split up into a large number of

rotational lines. The selection rule for the rotational quantum number J

is given by ∆J = J ′ − J ′′ = 0,±1. However, if Λ = 0, only ∆J = ±1 is

allowed. Therefore, the rotational transitions give rise to three series of lines

called P, Q and R branches corresponding to ∆J = −1, 0,+1 (Fig. 3.3).


Physical Aspects 57

The rotational lines of a given vibrational band cannot in general be re-

solved in low resolution spectra and therefore it results in a blended feature.

It can be resolved with a higher spectral resolution. For some molecules

the bands arising for the same change in the vibrational quantum number

in going from the upper to the lower electronic state have wavelengths very

close to each other. Therefore these bands cannot be resolved and give rise

to blended features known as band-sequences (Fig. 3.3). For example, ∆v =

0 of the Swan band sequence for the C2 molecule occurs around λ = 5165 A

(Fig. 4.1). To separate the bands from the band-sequences, it is necessary

to go in for even higher resolutions. To resolve rotational structure, still

higher resolutions are required (Fig. 5.17).

As in the atomic case, the molecular term can be written as

Z(2S+1)ΛΩ

where Z represents the designation of the electronic state, (2S + 1) the

multiplicity where S is the electron spin, Ω the total angular momentum

and Λ is the kind of term like Σ,Π, etc. As a typical case, the designation

of some of the band systems are given below for illustrative purposes.

(B2Σ−X2Σ) of the CN system and (A2Π−X2Σ) of the CO+ system

and so on.

3.5.3. Chemical subgroups

Certain chemical subgroups of atoms in a complex molecule, such as

-CH2- (methylene), -CH3 (methyl) etc. vibrate at characteristic stretching

and bending frequencies for that group. This gives rise to absorption or

emission lines at or near the same frequency regardless of the structure

of the rest of the molecule. Therefore infrared spectra can be used as a

diagnostic of the chemical subgroup present in the material. The chem-

istry dominated by the cosmically abundant H, C, N and O atoms and

the diagnostic bands fall almost exclusively in the near and mid-infrared

region between 2.5 and 25 µm. The various spectral signatures in the mid-

infrared region arising out of subgroups composed of H, C, N and O is

shown in Fig. 3.4. Figure 3.4 show that hydrocarbons have characteristic

frequency in the 3–3.4 µm and 5–8 µm regions depending on the exact

chemical composition and configuration of the larger molecules containing

the spectrally active groups. Therefore, infrared spectra is used extensively

to classify species and subgroup by chemical type, like aliphatic vs aro-

matic hydrocarbons etc. As for example, the 3.4 µm emission feature seen



Fig. 3.4 The vibrational frequency ranges of various molecular groups (Allamandola,L.J. 1984. In Galactic and Extragalactic Infrared Spectroscopy, Eds. M.F. Kessler and

J.P. Phillips, D. Reidel Publishing Company, p. 5: with kind permission of Springer

Science and Business Media).

from astronomical sources is generally attributed to C-H stretching modes

of -CH2 - and -CH3 groups in aliphatic hydrocarbons. This feature will be

seen in all biological molecules.

3.6. Isotopic Effect

The diatomic molecules which have the same atomic number but have

slightly different masses can give rise to a separation of the lines. This

is known as isotopic shift. The difference in the emitted frequencies be-

tween the two isotopic molecules arises through the difference in the reduced

masses of the two molecules. The spectroscopic constants of the isotopic

molecules (i) and the ordinary molecule are related by the following rela-


Physical Aspects 59

tions.

we(i) = ρwe

wexe(i) = ρ2wexe

and

weye(i) = ρ3weye (3.15)

where

ρ =

[µ

µ(i)

] 12

Here µ denotes the reduced mass of the molecule. The amount of the shift

depends upon the ratio of the masses and on the value of the vibrational

quantum number v. In the case when only the first term in the energy term

included, the isotopic shift is given by

∆νshift = (1− ρ)

[(v′ +

1

2

)ω′e −

(v′′ +

1

2

)ω′′e

](3.16)

Therefore, the isotopic shift is a function of the value of (1−ρ). The larger

the deviation from unity, the greater will be the shift of the lines. The sign

of (1− ρ) indicates the type of shift expected. The negative sign implies a

shift towards shorter wavelengths. For example, the (1,0) band of the blue

degraded Swan ∆v = +1 sequence occurs around 4737 A while that of12C 13C occurs around 4745 A.

The isotopic mass difference also has an effect on the rotational constant

B which in turn changes the rotational levels as

F i(J) = ρ2BJ(J + 1). (3.17)

Therefore, the isotopic effect could also be seen in the rotational spectra of

the molecules.

3.7. Franck-Condon Factors

The electronic transitions in a molecule give rise to several types of inten-

sity patterns depending on the type of the molecule. For example, for some

molecules (0, 0) transitions is the strongest, while for others the strongest

line may be for a different value (v′, v′′). The observed variations in the

intensity distributions can be understood in terms of the Franck-Condon

principle. The basic idea is the following. The electron jump takes place



from one electronic state to another preferentially at the turning points

of any vibrational level. This is due to the fact that the time of passage

between these two turning points is much shorter than the time spent at

the turning points. Therefore, the relative position and velocity after the

transition is the same as before the transition. Hence, if a transition takes

place from a given value of v′ of the upper state, the quantum number of the

jumped lower state depends on the location and shape of the two potential

curves. Figure 3.5 shows the expected results for two cases. In one case

Fig. 3.5 Understanding of the expected intensity distribution in emission according to

the Franck-Condon principle (see text). The minimum of the curve corresponds to theequilibrium position.

the minimum of the potential curves is nearly one above the other, while in

the second case, one is shifted for higher values of r. Based on the Franck-

Condon principle the (0, 0) transition should be the strongest for case 1.

For case 2, the strongest lines will be arising from the higher values of v′.

For emission lines, there will be two values of v′′, corresponding to points

B and D in Fig. 3.5 for which the intensity will be maximum. The locus of

the strongest bands is a parabola called the Condon parabola. The Condon

parabola can be calculated theoretically provided the potential curves are

known. The Franck-Condon factors give a measure of the relative band

intensities for an electronic transition.


Physical Aspects 61

3.8. Intensity of Emitted Lines

The intensity of a spectral line in emission, defined in ergs/sec, is given

by

I21 = N2A21hν21. (3.18)

Here N2 is the number of the species in state 2 and ν21 is the frequency of

the emitted radiation from state 2 to state 1. A21 is the Einstein coefficient

which gives the probability for a spontaneous transition from state 2 to

state 1, even without any external influence. The Einstein coefficient may

be expressed in terms of a parameter called the strength S21 of a line as

g2A21 =64π4ν3

21

3hc3S21 (3.19)

where g2 is the statistical weight or the degeneracy of the upper level. Other

quantities have their usual meanings. The mean life of state 2 is

τ2 =1

A21. (3.20)

It is instructive to see the order of value of τ for various types of transitions.

For electronic transitions τ ∼ 10−8 sec (A = 108 sec−1). For vibrational

transitions τ ∼ 10−3 sec(A = 103 sec−1) and for rotational transitions

τ ∼ 1 sec (A = 1 sec−1). The Einstein values B12 and B21 represent the

probability for the absorption and induced emission processes and is given

by

g2B21 = g1B12 =8π3

3h2S21. (3.21)

The total line strength S21 of the molecular line is the product of electronic,

vibrational and rotational components. Therefore

S21 = Se1SvibSrot. (3.22)

In general the probability of a transition between two states of eigenfunc-

tions ψ′ and ψ′′ is given by the equation

R =

∫ψ′µψ′′dr (3.23)

where µ is the dipole moment. R2 is proportional to the transition proba-

bility. Usually Sel and Svib are combined together as

(Se1Svib) ≡ Svibe1 =

∣∣∣ ∫ ψv′Reψv′′dr∣∣∣2 (3.24)



Here Re is called the electronic transition moment. ψv′ and ψv′′ are the

eigenfunctions for the vibrational states v′ and v′′. Re itself is defined by

the expression

Re =

∫ψ′eµeψ

′′e dτe (3.25)

where ψe’s are the electronic wavefunctions and µe refers to the electric

dipole moment for the electrons. In general the electron wavefunction ψealso depends to some extent on the internuclear distance r. Hence Reshould also depend on r. However, since the variation of Re with r is slow,

this variation is often neglected and Re is replaced by an average value of

Re. Therefore Eq. (3.24) becomes

Svibel = R2

e(rv′v′′)∣∣∣ ∫ ψv′ψv′′dr

∣∣∣2 (3.26)

where rv′v′′ is called the r- centroid and it is a characteristic internuclear

separation which can be associated with a given band (v′, v′′) and is given

by

rvv′′ =

∫ψv′rψv′′dr∫ψv′ψv′′dr

. (3.27)

The integral over the products of the vibrational wavefunctions of the two

states of Eq. (3.26) is known as the overlap integral and is generally called

the Franck-Condon factors of the (v′, v′′) band. Therefore, Eq. (3.26) can

be written as

Svibel = R2

e(rv′v′′)qv′v′′ (3.28)

where

qv′v′′ =∣∣∣ ∫ ψv′ψv′′dr

∣∣∣2.Hence Eq. (3.22) becomes

S21 = R2e(rv′v′′)qv′v′′Srot. (3.29)

Hence the electronic transition moment R2e refers to r-centroid rv′v′′ and

Srot is usually known as Holn-London factors. Equation (3.29) shows that

the total strength of a molecular line is essentially given by the product of

the three-strength factors, namely the electronic transition, Franck-Condon

factors and the Holn-London factors.

The values of the electronic transition moment for any band basically

has to come from the laboratory measurements of the intensity of the lines,

as theoretical calculations are very difficult. Enormous amount of work has


Physical Aspects 63

been carried out in various laboratories over the world to extract this basic

data from the line intensity measurements of various bands. But the data

is still very meagre. So for most of the cases, one ends up using either the

mean value or an approximate value for the electronic transition moment.

The Franck-Condon factors can be calculated from a knowledge of the

wavefunction of the vibrational levels, which comes out of the solution of

the Schrodinger equation. For this, the potential function U(r) has to be

expressed in a convenient and mathematical form. There are various repre-

sentations of the potential curve. The well-known function is that of Morse

and is generally called Morse function. This has been extensively used in

the literature as it is quite simple and convenient. However it does not

represent the potential curves for all the cases exactly. So other expres-

sions for the potential have been suggested. The one that is commonly

used in recent years is known as the RKR potential referring to the authors

Rydberg, Klein and Rees, who proposed it. In this method, the potential

curve is constructed point by point from the laboratory measured values

of the vibrational and rotational levels. The potential curves obtained in

this way are much superior in many cases compared to Morse potential

representation. The advantage of this method is the fact that it uses exper-

imentally determined values of the quantities. However, the disadvantage

is that the experimental values are often not of good quality and, in addi-

ton, this method is quite cumbersome. Many people have written computer

programs to evaluate qv′v′′ and r -centroids for given values of input pa-

rameters. For many molecules of astrophysical interest, the values of qv′v′′

have been published in the literature.

On the other hand, the rotational strength factors have to be calculated

from the theory. They depend upon the structure of the molecule, type

of coupling, type of transition involved, etc. Until recently there existed

a lot of confusion in the definition and normalization of Holn-London fac-

tors. This has been clarified by several workers. The expression for the

Holn-London factors has been evaluated for various cases of interest and is

available in the literature. Computer programs have also been written to

compute these factors.

It is also possible to get an estimate of the Einstein A value or the

oscillator strength f , directly from the laboratory measurements of the

intensity of lines. Various techniques and methods have been employed to

measure these quantities. The measurements are however hard to make and

so there are not many measurements available at the present time. Even if

they are available, the values by different methods or by different observers



often disagree. Therefore, one has to make use of laboratory measured

values along with the calculated quantities in any particular situation.

3.9. Boltzmann Distribution

The calculation of the intensities of lines involves a knowledge of the

relative population of atoms or molecules in different excited states. In

a thermal equilibrium in which every process is balanced by its inverse,

the population distribution among various levels is described according

to Boltzmann formula. The population distribution between two discrete

states 1 and 2 with an energy separation ε12 = E2 − E1 is given by the

expression

n2

n1=g2

g1e−ε12/kT (3.30)

where g1 and g2 are the statistical weights of the two levels and for an atom

it is equal to

g(J) = 2J + 1

where J is the total angular momentum quantum number. The expression

can be used to calculate the population distribution in different levels for a

given temperature. Conversely, with a knowledge of the population in two

or more levels, the excitation temperature can be determined.

3.10. Λ-Doubling

The assumption that the interaction between the angular momentum

due to rotation of the molecule as a whole and the electron orbital angular

momentum can be neglected is valid only for the case of Λ = 0. This

corresponds to the case of Σ states. However, for the electronic states

Π,∆, · · · , for which Λ 6= 0, this interaction splits the rotational energy

levels. This is called Λ-doubling. The splitting of the levels is generally

very small and is of the order of 1cm−1 or less. Therefore the transitions

between these levels will give rise to lines which lie in the radio frequency

region. For transitions of the type (1Σ −1 Σ) there is no Λ-doubling. But

for others like (2Σ−2 Π), the effect of Λ-doubling can be seen. Λ-doubling

is the characteristic feature of molecules like OH,CH, etc., which have large

rotational constants. The OH radical has been studied more extensively

than any other radical. The energy level diagram of the hydroxyl radical,


Physical Aspects 65

with only a few vibrational levels is shown in Fig. 3.6. The strong (0,0) band

Fig. 3.6 Energy level diagram of the OH molecule and the associated Λ-doubling.

which is generally seen in emission in comets arises from the transitions as

shown in Fig. 3.6. The lowest rotational level, corresponding to v′′ = 0

and J ′′ = 3/2 is further expanded in Fig. 3.6 to show the Λ-splitting. The

two Λ-doubling states are further split by the hyperfine interaction with the

hydrogen nucleus. Therefore in all there are a total of four lines (Table 3.1).

All these lines have been seen from many astronomical sources.

Table 3.1 Radio lines of the OH molecule.

Frequency (MHz) Relative Intensities Transition

of lines probability (sec−1)

1612.23 1 4.50 (-12)1665.40 5 2.47 (-11)

1667.36 9 2.66 (-11)

1720.53 1 3.24 (-12)

3.11. Photochemistry of Water

Extensive studies pertaining to photochemistry of water has been car-

ried out through laboratory and theoretical studies. All these studies have

shown that there are a large number of channels through which water can

go through depending upon the impinging energy of the photon. The re-

sulting fragments can further divide in several ways. All these possible



pathways are given in Table 3.2.

Table 3.2 Water photodissociation products.

Reaction Threshold

wavelength (A)

(a) H2O + hν → OH+ H 2424.6

→ OH(A2Σ+) + H 1357.1

→ H2 + O(1D) 1770→ H2 + O(1S) 1450

→ H + H + O(3P) 1304→ H2O+ + e 984

→ H + OH+ + e 684.4

→ H2 + O+ + e 664.4→ OH + H+ + e 662.3

(b) OH + hν → O + H 2823.0

→ OH+ + e 928

H2 + hν → H + H 844.79

→ H+2 + e 803.67

→ H + H+ + e 685.8O + hν → O+ + e 910.44

H + hν → H+ + e 911.75

(a) Main pathways. (b) Fragment pathways. Adapted from

Feldman, P.D., Cochran, A.L. and Combi, M.R. 2005, In

Comets II, Eds. M.C. Festou, H.U. Keller and H.A. Weaver,Univ. Arizona Press, Tucson, p. 425.

In the first reaction the product OH(A2Σ+) is produced in the highly

excited state which decays giving rise to “prompt” emission lines. In the

third reaction, the product is in the metastable levels O(1D) and O(1S)

which give rise to forbidden lines.

These results are of direct relevance to comets as water is the dominant

molecular specie in the coma of comets at heliocentric distances of around

1AU. In fact most of the products as mentioned in Table 3.2 have been

observed in comets through spectroscopic means.

Among all the photo-dissociation fragments of H2O, the velocity dis-

tribution of H in the reaction producing the products of H and OH is

completely dominated by non-thermal component. This arises due to the

fact that there is excess energy available over and above the energy required

to overcome the chemical bond energy. This excess energy is imparted to

the H atoms which result in the velocity of ∼ 17.8 km/sec. OH radical has


Physical Aspects 67

an excess velocity ∼ 1.05 km/sec.

3.12. Silicate

There are various types of silicates. They differ in their chemical com-

position and mineral structure. The basic building block of silicate material

is SiO4. They are tetrahedral in shape. In the tetrahedron, the four oxy-

gen atoms of SiO4 are located one at each corner, while Si atom is at its

centre. The two main forms of silicate material of intererst are Olivine

and Pyroxene (Fig. 3.7). Olivines can be considered as solid solutions of

Fig. 3.7 Different forms of Olivines and Pyroxenes and their structure (Henning, Th.

1999, In Asymptotic Giant Branch Stars, Eds. T. Le Bertre, A. Lebre and C. Waelkens,

IAU Symposium 191, ASP Conference Publications, p. 221:By the kind permission ofthe Astronomical Society of the Pacific Conference Series).



Mg2SiO4 and Fe2SiO4 , while Pyroxenes of MgSiO3 and FeSiO3. They can

be represented by a general formula as Mg2xFe2−2xSiO4 for olivine and MgxFe1−xSiO3 for pyroxenes. The value of x can take values between 1 and 0.

The end members of olivine class are Forsterite (x = 1, Mg2SiO4) and Fay-

alite (x = 0, Fe2SiO4). The corresponding ones for pyroxene are Enstatite

(x = 1, MgSiO3) and Ferrosilite (x = 0, FeSiO3). Pyroxene has chain-like

structure while olivine has island-like structure. In the case of amorphous

silicate, long range order is absent due to blending of structures and only

short-range order exists. The two crystalline silicates exhibit both short-

range and long-range order. The degree of long-range order (amorphous

to crystalline) has a considerable effect on the spectroscopic properties of

the material. For example, amorphous silicate exhibit broad smooth spec-

tral features, while crystalline silicates show sharp and distinct features.

Amorphous and crystalline silicates are present in cometary dust.

3.13. Annealing

Amorphous silicate is the most common form present in astronomi-

cal objects. In fact the dominant component of silicate in the interstellar

medium is amorphous silicate. Since crystalline silicate is present in comets,

it is important to understand how amorphous silicate can be transformed

to crystalline silicate.

The transformation of amorphous to crystalline state is possible by the

process of annealing in which the ordered arrangement of silicate tetrahe-

dra is brought about by atomic diffusion. In this process, thermal diffusion

brings about rearrangement of the structural units (tetrahedra) which re-

sults in long-range order. Diffusion in solids is a result of activation of

lattice defects. Therefore the poorly ordered state of the amorphous ma-

terial slowly drifts towards more energetically favourable positions leading

to macro-crystalline structure.

Several experiments have been carried out to see the actual change over

from the amorphous state to crystalline state. Magnesium silicate which

showed a broad feature at 9.3 µm to start with showed two features at 9.8

and 11.0 µm after annealing at 1027 K. These features and the spectral

feature at 20 µm showed the formation of olivine. The change over from

amorphous to crystalline can take anywhere between a few hours to a day

or so. In the change over process the band length or angle within the SiO4

tetrahedron is not altered but simply leads to ordered arrangement of the


Physical Aspects 69

SiO4 units. As an example the annealing of CaMgSi2O6 smoke is shown in

Fig. 3.8.

Fig. 3.8 Annealing of CaMgSi2O6 smoke at 1050K and for times from 1 to 40 hours.

The development of structures in the curve can clearly be seen. The curves for 1 to 40

hours are shifted vertically for clarity (Colangeli, L., Henning, Th., Brucato, J.R. et al.2003. Astron. Astrophys, Rev., 11, 97).

3.14. Carbon

Carbon can exist both in crystalline and amorphous form. Some of the

forms of carbon are graphite, diamond and amorphous carbon. Amorphous

carbon does not exhibit a long-range order in their arrangement. There are

serveral classes of materials of this category which depend mostly on their

appearance, for example, glassy carbon or soot. Aliphatic molecules have

open structures, while aromatic molecules have ring structures, like ben-

zene, C6H6. Benzene is the building block for a wide range of hydrocarbon

molecules. Molecules with several aromatic rings are called Polycyclic Aro-

matic Hydrocarbons (PAHs). PAHs are the most thermodynamically stable

hydrocarbon compounds that exist in the gaseous form. PAHs can exist in

neutral and ionized states. The size of PAHs could be around 25 to 100

carbon atoms. Therefore wide variety of PAHs can exist under astrophysi-

cal conditions including comets. PAHs possesses rich spectra with specific



spectral signature, a high photo-stability and also contain abundant carbon.

The mid infrared region (2 to 25 µm) is where the fundamental vibrational

modes of PAHs and their overtones are active. The C - H out-of-plane

bending feature occurs at around 11 µm, C-C stretching around 6 µm and

C-H in-plane bending around 8.7 µm. A series of emission lines at 3.3, 6.2,

7.7, 8.6 and 11.3 µm seen in astronomical objects is generally attributed to

PAHs. However no specific molecule has been identified.

3.15. Solar Radiation

The subject of the study of the total amount of energy as well as the

spectral distribution in the solar spectrum is an old one. It is of great

intererst in various fields. It has therefore been studied extensively in the

entire range of the electromagnetic spectrum. The detailed and extensive

measurements are carried out in the case of the Sun because of its proximity,

it being a typical main sequence star and also because of its effects on the

Earth’s environment. It also turns out that most of the phenomena that one

observes from a comet is associated directly or indirectly with the radiation

field of the Sun.

The surface temperature of the Sun is about 6000K. According to

Planck’s distribution law [Eq. (3.1)] most of the energy is concentrated

in the visible region of the spectrum. Therefore the amount of energy mea-

sured from the Earth corrected for the atmospheric transmission gives a

good estimate of the total energy of radiation. Of course, one should try

to allow for the UV and IR radiation which is cut off by the Earth’s at-

mosphere. The intensity of the solar radiation as seen from the Earth’s

atmosphere at the Earth’s mean distance from the Sun is called the solar

constant. To get the solar constant value, extensive ground based measure-

ments have been carried out. The average value for the solar constant is

found to be ∼ 1.95 cal/cm2/min.

The spectral energy distribution in the visible region is quite smooth

and can be represented well by a black body of temperataure 6000K. Su-

perposed on it are the absorption lines of the gases of the solar atmosphere.

For measuring the solar emission in the UV region, it is necessary to go

above the Earth’s atmosphere. Such observations have been carried out us-

ing rockets and satellites. The solar flux decreases very rapidly to shorter

wavelength. As one goes towards the UV spectral region, the number of

emission lines arising out of different ionization levels of the various ele-


Physical Aspects 71

ments increases. In fact below around 1500 A, the major portion of the

energy is in emission lines rather than in the continuum (Fig. 3.9). As one

Fig. 3.9 A representative spectral distribution of the quiet Sun between 1150 to 1400 A

(Dupree, A.K. and Reeves, E.M. 1971. Ap. J. 165, 599).

goes towards shorter and shorter wavelength, one is essentially seeing the

higher and higher temperature regions, namely the chromosphere and the

corona of the Sun, from where the emission lines arise. Therefore the shape

of the continuum radiation in the UV and far UV regions depend to certain

extent on the resolution of the instrument. The higher the resolution of

the instrument the more the number of weak lines can be resolved. The

intense lines in the UV region are the Lyman α and Lyman β occuring at

1215.7 A and 1025.7 A, respectively. They have fluxes of about 3×1011 and



2× 109 photons/cm2/sec at the top of the atmosphere as compared to the

continuum level in the same region of about 108 photons/cm2/sec. Lyman

α and Lyman β emission lines are quite important as they can dissociate,

ionize or excite the species. There could be some uncertainty in the data in

the UV region. In addition, the emission could vary with the solar condi-

tions as for example, whether the Sun is quiet or flares are present and so

on. The solar UV flux also varies considerably with 27 day solar rotation

period. It also varies with 11 year solar activity cycle which can cause the

solar flux to vary by a factor of 2 to 4 at wavelength shorter of 1000 A.

It will become clear later on that the emission and absorption lines,

strong as well as weak present in the solar spectrum, as well as their vari-

ations play a crucial role in the formation of spectral line in a cometary

atmosphere.

3.16. Solar Wind

The solar plasma from the corona that flows out continuously into the

interplanetary medium is called the solar wind. The possible existence of

such plasma flows came from the pioneering investigations of Biermann

based on the study of plasma tail of comets. The theoretical work of

Parker in the 1950’s based on the hydrodynamic expansion of the solar

corona predicted the expected nature of the solar wind. The theory also

showed that the magnetic field originating in the photosphere and dragged

into the interplanetary medium takes the form of an Archimedean spiral

structure. The observations carried out with satellites later on confirmed

Parker’s theory of the solar wind. The actual interplanetary magnetic field

measurements have established the Archimedean spiral structure for the

magentic field. The mean values obtained from observations for some of

the physical paramters of interest for the solar wind near the Earth are

the following: number density of electrons or protons, Ne ∼ Np ≈ 5/cm3;

velocity ≈ 450 km/sec; electron temperature ∼ 1.5× 105K; magnetic field

≈ 2.5γ where γ = 10−5 gauss. However, it is well known that the Sun is not

quiet all the time and lot of activities of violent nature do take place on the

surface of the Sun, like flares, bursts, etc. These have a great effect on many

of the geomagnetic activities. In fact there is an almost one-to-one relation

between the disturbance occuring on the Sun and the effect observed on

the terrestrial atmosphere. The existence of such a correlation has been

known since early times. These effects arise basically due to the fact that a


Physical Aspects 73

disturbance on the surface of the Sun releases suddenly a large amount of

plasma which is coupled through interplanetary magnetic field. Therefore,

the average physical properties of the solar wind change with the solar con-

ditions. During disturbed periods the solar wind velocity and the density

at 1 AU could vary by a factor of two or so. The magnitude as well as the

direction of the magnetic field also change. The extensive measurements

made with satellites have given an enormous amount of information with

regard to the variation of physical quantities with heliocentric distance for

quiet as well as for disturbed solar conditions. These measurements have

also shown the presence of magnetic sector boundaries in the magnetic field

which have opposite polarity. These regions are separated by null surfaces

where the magnetic field is zero. Since the plasma tail of a comet and

the solar wind are coupled to each other through interplanetary magnetic

fields (Chap. 10), the interpretation of the observed cometary features re-

quires a knowledge of the exact conditions of the solar wind at the time of

observation.

Problems

1. Derive Eq. (3.1) in frequency units.

2. Calculate the wavelength of vibrational bands of the Swan system (d3Π

- a3Π) for the C2 molecule and (B−X) transitions of the H2 molecule

for v′′=0,1,2 and v′=0,1. This will help in understanding the meaning

of band sequence. What instrumental resolution is required to resolve

vibrational and rotational structure of the Swan bands.

3. Derive the expression for the isotopic shift for the vibrational and ro-

tational transitions including second order terms.

4. Calculate the isotopic shift between 12C12C and 12C13C for the (1,0)

band of the Swan system.

5. Calculate the frequency and wavelength of the line arising out of the

transition between n = 109 to 108 in hydrogen and carbon atoms.

Discuss its significance.

6. Is there a Doppler effect when the observer or the source moves at

right angles to the line joining them? How can the Doppler effect be

determined when the motion has a component at right angles to the

line?

7. Make an estimate of the temperature of the Sun from the fact that it

can be seen.



8. Explain why the spectrum of the Sun at wavelengths less than 1500 A

consists entirely of emission lines.

9. What evidence do we have, direct and indirect, of the existence and

properties of solar wind?

10. What is the probable mechanism responsible for the generation of the

solar wind?

11. Sun’s radiation striking the Earth has an intensity of 1400

watts/meter2. Assuming the Earth is a flat disk at right angles to

the Sun’s rays and that the incident energy is completely absorbed,

calculate the radiation force acting on the earth. Compare it with the

force due to the Sun’s gravitational attraction.

12. Calculate the Einstein A-value for the rotational transitions 2 to 1 in the

ground electronic state and compare it with the value for the electronic

transitions between A and X levels for the CO molecule.

References

1. Herzberg, G. 1950. Molecular Spectra and Molecular Structure 1. Spec-

tra of Diatomic Molecules, New York: Van Nostrand Reinhold Com-

pany.


CHAPTER 4

Spectra

The main goals of the study of spectra are to identify the species re-

sponsible for the observed lines and to obtain information about their

abundances and the physical conditions present in the source. It is no

exaggeration to say that our present understanding about various aspects

of cometary phenomena has come directly or indirectly from the study of

their spectra.

The first photographic observations in the visible region was carried

out in 1864. This was gradually superceded by photoelectric detectors of

various kinds followed by charged coupled devices (CCDs). Therefore it

is possible to make high spectral resolution observations with the existing

large size optical telescopes.

The observations in the UV spectral region below 3000 A was carried out

first with rockets and then by Orbiting Astronomical observatory (OAO-2)

in 1970. This was followed by the launching of International Ultraviolet

Explorer (IUE) in 1978, Hubble Space Telescope (HST) in 1990 and Far

Ultraviolet Spectroscopic Explorer (FUSE) in 1999. These satellites made

it possible to explore the spectral region from around 3000 A down to

about 900 A. The observations carried out with Rontgen Satellite (ROSAT),

Extreme Ultraviolet Explorer (EUVE) and Chandra X-ray observatory have

extended the wavelength limit to X-ray region of around 1 A.

In the near infrared region, observations have been carried out with

balloons and with the Keck telescope, Infrared Telescope Facility (IRTF),

Subaru telescope etc. at Mauna Kea in Hawai with sensitive detectors.

Kuiper Airborne observatory (KAO) was very helpful in making observa-

tions in the infrared. Infrared Space Observatory (ISO) made it possible

to study far−infrared spectral region with high resolving power. This has

been followed by Spitzer Space Observatory (SSO) which has opened up

75



new avenue for making infrared observations.

The availability of large size radio telescopes has made it possible to

study molecules in the millimetre and radio wavelength regions.

4.1. Main Characteristics

The identification of the spectra of comets is one of the active areas of

study. This is due to the combination of many factors such as better in-

struments, availability of space vehicles, computation techniques, etc. The

identification of the lines in the spectra of a comet is quite a complex and

difficult task. The usual procedure is to look for the coincidences between

the laboratory wavelength of lines and the observed wavelenghts in the spec-

tra of a comet which have been corrected for the velocity effect (Chap. 3).

With this method many of the well−known atoms and molecules which are

seen in other astronomical objects have been identified in the spectra of

comets. With better resolution as well as going into new spectral regions,

many more new species have been identified.

The extensive spectral observations carried out on Comet Halley which

have been compiled in the Atlas of Cometary Spectra, show the vast extent

of the lines which could not be identified. There are also large number of

unindentified lines in the high dispersion (λ/∆λ = 60,000) visible spectrum

of Comet 122P/de Vico. Therefore much work remains to be done in the

area of identification of lines. It is appropriate at this point to mention

the fact that the problem of identification of lines is intimately connected

with the availability of spectroscopic data pertaining to various atoms and

molecules. In addition, the astrophysical environment is vastly different

from those of laboratory conditions. Therefore many of the lines that may

be present in the astronomical spectra cannot be produced under labora-

tory conditions. Hence laboratory data is not available for each and every

transition for all the atoms and molecules. Therefore, there could always

be lines in the astronomical spectra which cannot be identified. It is inter-

esting to note that several transitions were observed first in the cometary

spectra before being studied in the laboratory. The well−known case is

the bands of the ion CO+, generally called the Comet−Tail System. Other

examples are the bands of C3 and H2O+.

The observations carried out in the visual spectral region of around

3000 to 8000 A have been the main source of information for the study

of spectra of cometary atmosphere for the last several decades. These


Spectra 77

studies had limitations due to photographic techniques and also most of the

spectroscopic observations could be made only on bright comets at small

heliocentric distances. Early observations were also limited to spectra taken

at low spectral resolution. In recent years many of these limitations have

been overcome. Based on the spectra in the visual region, it is possible to

arrive at some general pattern regarding the main characteristic features of

the spectra of comets.

At large heliocentric distances, the comet appears as a point source.

This is due to the reflection of the solar radiation by the nucleus. For

heliocentric distances, r >∼ 3 AU, the spectrum is mainly the continuum

radiation arising mostly due to the scattered solar radiation by the dust

particles present in the cometary atmosphere. The emission lines of the

various molecules appear roughly in a sequence as the comet approaches

the Sun. The molecular bands first to appear are those of CN at r ∼ 3

AU followed by the emission from C3 and NH2(r ∼ 2 AU). Thereafter at

r ∼ 1.5 AU the emission from C2 (Swan bands), CH, OH and NH appear

in the spectrum. They are often strong enough to reveal their structure.

At r< 1.5 AU emission from CO+, OH+, N+2 and CH+ appear. Emission

from Na appear around 0.8 AU. The relative intensity of emission bands

and continuum varies from comet to comet. In sun-grazing comets, say for

r ∼ 0.1 AU, numerous metallic emission lines crowd the spectrum.

The spectrum of a plasma tail shows mostly the presence of ionized

species and generally at r< 2 AU. The notable exceptions include the Comet

Humason (1962 VIII) where CO+ was seen even at r> 6 AU. Among the

observed ions in the plasma tail of a comet, the emission from CO+ domi-

nate.

As a typical case, Fig. 4.1 shows the spectra of Comet Encke in the

visual region taken at a resolution (full width at half maximum intensity)

of ∆λ = 7 A. The Swan band sequences corresponding to ∆v = −1, 0

and +1 of the C2 molecule (d3II - a3 II), whose wavelengths lie around

5635, 5165 and 4737 A respectively, are the strongest in the spectra. The

vibrational structure could be resolved with better resolution spectra. The

resolution of the rotational structure requires still higher resolution. In fact

it has been possible to resolve completely the rotational structure.

The spectra of Comet Halley taken at a spectral resolution of 0.07 A

beautifully shows the rotational structure of the (0, 0) Swan band of the

C2 molecule(Fig. 5.17). The spectra taken in recent times by instruments

with very high resolving power, λ/∆λ ≈ 60,000, shows in great detail, the

rotational structure of individual bands. Since the Swan bands of the C2



Fig. 4.1 Typical scanner spectrum of Comet Encke using the 3−m Lick Observatory

Telescope in the visual region. Courtesy of Spinrad.

molecule dominate the spectrum in the visual region, to a first approxima-

tion, it also determines the ‘visual diameter’ of the head of the comet. It

may also be noted that the wavelength of ∆v = +1 band sequence of the

isotopic molecule 12C13C (4745 A), is shifted by about 7 A with respect

to that of 12C12C (4737 A) and it has been seen in the spectra of many

comets. Unfortunately this isotopic feature is blended strongly with the

emission from the NH2 molecule ((A2A1− X2B1) system). Other bands of

NH2 can also be seen. Several other isotopic lines have been seen.

The emission due to C3 molecule has a broad feature extending roughly

from 3950 to 4140 A, with a strong peak around 4050 A. The identification

of C3 feature in comets was difficult as the laboratory analysis was not

available. Laboratory studies have confirmed the observed lines as due to

C3 molecule and it has a complex spectra. Until its indentification, this

broad feature was simply known as ‘4050 A’ Group. Various transitions

of the CN molecule, both the red (λ ∼ 7800 A −1 µm) and the violet

(λ ∼ 3600 −4200 A) systems corresponding to transitions (A2Π−X2Σ+)

and (B2Σ+−X2Σ+) have been identified. The violet system of CN is one of

the prominent feature in the spectra of comets. The rotational structure of

CN(0, 0) band is well−resolved (Fig. 5.1). High spectral resolution obser-

vations of the violet system of CN molecule resolve completely the isotopic

feature which has been used to derive 12C/13C ratio. For CH molecule, both

the systems (A2∆−X2Π) and (B2Σ−X2Π) have been seen. The wavelengh

of (0, 0) band of (A − X) is at 4314 A and that of (B − X) is at 3886 A. The

(A−X) band is weaker than the (B−X) band. The (0, 0) band of (A−X)

transitions has been observed at high resolution.


Spectra 79

The lines of (0, 0) band of the NH molecule arising out of (A3Πi−X3Σ)

transitions lie in the region of 3345 and 3375 A. The (0, 0) and (1, 1) bands

corresponding to (A2Σ+−X2Π) of OH at 3090 A and 3140 A, which is close

to the atmospheric cut−off were detected from ground−based observations.

The rotational structure was partially resolved. Numerous lines belonging

to (A3Π−X3Σ) system of OH+ (λ ∼ 3500 A) have been identified. The lines

of H2O+ (λ ∼ 5500 − 7500 A) were identified for the first time in Comet

Kohoutek. The bands of (A2Π−X2Σ) of CO+ around λ ∼ (3400−6300 A)

has been seen. The lines of CH+ seen in a cometary spectra is the (0, 0)

band at 4230 A belonging to (A1Π−X1Σ) transitions. The (B2Σ+u−X1Σ+

g )

(0, 0) band of N+2 at 3914A has been seen(See Sec. 6.2.7).

The lines arising out of metals, K, Ca, Fe, V, Cr, Mn, Ni and Cu have

been seen in the sun-grazing Comet Ikeya−Seki. The sodium D lines at

5890 and 5896 A also show up for r <∼ 1.4 AU.

The very first lines emitted by OH following the dissociation of H2O

called ‘Prompt emission’ lines have been seen in UV (∼ 3100 A) (Sec. 5.1.5)

and infrared (∼ 3 µm) regions (see Sec. 6.2.2.)

The use of rockets and satellites has made it possible to extend the

observations into the ultraviolet region. This is the region of the spectrum

say from 1000 to 4000 A where the abundant atomic and molecular species

have their resonance transitions. In fact the spectra of comets taken in

the ultraviolet region has clearly demonstrated the richness of molecular

emissions in this spectral region. Comet West in 1976 provided a good

opportunity to secure high quality spectra in the UV region as the comet

was quite bright. For illustration, the rocket spectra of this comet covering

the wavelength region from 1600 to 4000 A is shown in Fig. 4.2. Many

molecules like CS, CN+ and others were identified for the first time based

on the spectra of Fig. 4.2 and others. The wavelength of the (0, 0) band of

CS arising out of the electronic transition (A1Π−X 1Σ+) is at 2576 A. The

wavelength of the (0, 0) band of CN+ arising out of (1Σ−1 Σ) transition is

at 3185 A. The (0, 0) transition, (A−X) of OH at 3090 A which was weak

when observed from the ground is found to be very strong when observed

outside the Earth’s atmosphere. The rotational structure of OH bands has

also been resolved completely.

The strong emission (A1Π−X1Σ+) bands of CO commonly referred to

as the Fourth Positive system occuring around 1500 A region were also

identified for the first time in Comet West. Figure 4.3 shows the spectrum

covering this spectral range. The Cameron band system near 2050 A arising

out of dipole forbidden triplet and singlet electronic states (a3Π−X1Σ+) of



Fig. 4.2 Rocket spectrum of Comet West in the 1600 to 4000 A spectral region. The

richness of molecular emissions can be seen. Courtesy of Andy Smith.

Fig. 4.3 Low resolution spectrum of Comet West taken from an Aerobee rocket in the

wavelength region 1300 to 1700 A which shows bands of CO molecule (Feldman, P.D.and Brune, W.H. 1976. Ap. J, 209, L45).

CO was observed with HST for the first time in 1991 in Comet Hartley 2

(1991 XV). After its detection, it was also shown to be present in the earlier

data acquired with IUE in several others comets. Several other bands of

CO belonging to Hopfield−Birge system (B1Σ+−X1Σ+), (C1Σ+−X1Σ+)


Spectra 81

and (E1Π−X1Σ+), have been seen in the spectral region 1075 and 1155 A

obtained with FUSE (See Fig. 4.6). The presence of CO+2 ion in comets

came from the detection of the bands of (A2Π− X2Π) system in the region

3300−3900 A. The (B2Σ − X2Π) double band system whose wavelength

occurs around 2890 A has also been identified. In the region of 2000 to

2600 A the First Negative bands (B2Σ+−X2Σ+) of CO+ are generally very

strong. The Mulliken System of the C2 molecule corresponding to ∆v = 0

has a wavelength around 2300 A. This feature is present in Fig. 4.2. The

relevant spectra of Comet Bradfield is shown in Fig. 4.4. Several strong

Fig. 4.4 The ultraviolet spectrum of Comet Bradfield 1979 X obtained from the Inter-national Ultraviolet Explorer (Weaver, H.A., Feldman, P.D., Festou, M.C. and A’Hearn,

M. F. 1981. Ap. J, 251, 809).

emission bands of (B3Σu−X3Σg) of the S2 molecule have been identified

in the wavelength region of 2800−3100 A based on the beautiful spectra of

the Comet IRAS−Araki−Alcock (1983VII) as shown in Fig. 4.5. The P1

lines of H2 Lyman system (B1Σ+− X1Σ+g ) were detected for the first time

in Comet C/2001A2 (LINEAR) with FUSE as shown in Fig. 4.6.

The earlier reported detection of lines of N+2 in the low resolution spectra

of comets are not seen in the high spectral resolution observations of some

recent comets (See Sec. 6.2.7).

The dissociation of the molecular species should finally lead to their

constituent atoms such as H, O, C, N and S. These lines lie in the UV

region. Shortward of 1800 A, the spectrum is dominated by multiplets of

CI at 1561 A, 1657 A, OI at 1304 A and CII at 1335 A. Many weak CI



Fig. 4.5 The spectrum of Comet IRAS−Araki−Alcock in the wavelength region 2500

to 3200 A obtained with the International Ultraviolet Explorer Spacecraft. The imageNo. LWR 15908 was taken on May 11, 1983 with the nucleus centred, when the comet

was 0.032 AU from the Earth. The new features corresponding to (B−X) system of S2

are identified (A’Hearn, M.F., Feldman, P.D. and Schleicher, D.G. 1983. Ap. J., 274,L99).

features are also present. The lines of SI multiplet at 1807, 1820 and 1826 A

were detected for the first time in Comet West. (Figs. 4.2 and 4.4). Two

other multiplets of S are sometimes seen at 1429 and 1479 A.

Some of the molecules detected through their spectral characteristics

are given in Table 4.1.

The observations made with the orbiting astronomical observatory

(OAO−2) satellite in 1970 on Comet Kohoutek and on other comets in

the light of the hydrogen Lyman α line at 1216 A led to the discovery of a

hydrogen halo around the visible coma (Fig. 1.13). This important obser-

vation also led to the realization that the mass loss rates from comets are

much higher than previous estimates which were based on observations in

the visual spectral region.

The first X−ray detection from a comet was from Comet Hyakutake in

1996 (Fig. 10.15). The X−ray spectra obtained on Comet LINEAR from

Chandra X−ray observatory in the energy range 0.2 to 0.8 KeV showed that


Spectra 83

Fig. 4.6 The lines of H2 seen in the FUSE spectrum of Comet C/2001 A2 (LINEAR)

on July 12, 2001. (Feldman, P.D., Weaver, H.A. and Burgh, E.B. 2002. AP.J., 576,

L91, Reproduced by Permission of the AAS).

the spectra was dominated by the emission arising out of highly charged

ions of C, N and O (Sec. 10.7).

The extension of observations into the infrared region is important as

this spectral region is characterized by vibration−rotation transitions of

many molecules. Some of the bands of the CN red system (A2Π−X2Σ)

occurring around λ ∼ 8000 A were first seen in the Comet Mrkos (1957 V)

based on photographic spectra. This has later been observed from many

comets. Several other bands of CN can also be seen. Many weak features

of NH2 and the Phillips bands of C2 are present. High resolution spectra

has resolved the rotational structure.

At wavelengths larger than 1 µm, lines of several molecules have been

detected. In particular the first and direct detection of H2O in Comet

Halley came from the high resolution observation of ν3 vibrational band at

2.7 µm with the Kuiper Airborne Observatory (Fig. 4.7). In the spectral

region around 2.9 µm multiple nonresonant fluorescent lines of H2O called

hot bands have been seen.



Table 4.1 Some of the observed spectroscopic

features of molecules.

Molecule Transition Wavelength

(A) (0,0) band

OH A 2Σ+− X 2Πi 3085

CN B 2Σ+−X 2Σ+ 3883

A 2Π−X 2Σ+ 7873a

C2 d3Πg− a 3Πu 5165

A 1Πu− X 1Σ+g 7715b

D1Σ+u−X 1Σ+

g 2313

C3 A1Πu−X 1Σ+g 3950-4140∗

CH A 2∆−X 2Π 4314B 2Σ−X 2Π 3886

CS A 1Π−X 1Σ+ 2576

CO A 1Πu− X1Σ+ 1300-1800∗

a 3Πr− X 1Σ+ 1900-2200∗

B 1Σ+−X1Σ+ 1150.5

C 1Σ+−X 1Σ+ 1087.9E 1Π−X 1Σ+ 1076.1

NH A 3Πi− X3Σ− 3360

NH2 A2A1−X2B1 4500-7350∗

H2 B 1Σ1−X 1Σ+g 1071.6,1166.8c

S2 B3Σu−X3Σg 2800-3050∗

CH+ A1Π− X 1Σ+ 4230

OH+ A3Π−X 3Σ− 3565

CO+ A2Π−X 2Σ+ 4273a

B 2Σ+−X 2Σ+ 2190

N+2 B2Σ+− X2Σ+ 3914

H2O+ A2A1−X2B1 4270-7540∗

CO+2 A2Πu−X 2Πg 2800-5000∗

B2Σu−X2Πg 2890

*Range of wavelength.

a. (2,0), b. (3,0), c. (6,1) and (6,3)

Feldman, P.D. et al. 2002, Ap. J., 576, L91.Weaver, H. 1998, Comets, In The Scientific Im-

pact of the Goddard High Resolution spectro-

graph, eds. J.C. Brandt et al. ASP Conf. se-ries No. 143, p. 213.Feldman et al. 2005, In Comets, eds. M.C.

Festou, H.U.Keller and H.A. Weaver, Univ.Arizona Press, Tucson, p. 425.

In the 3 µm region strong emissions from hydrocarbons are present.

This is shown in Fig. 4.8 where the emissions from CH4 (Methane), C2H2

(Acetylene) and C2H6 (Ethane) can clearly be seen. A broad emission

feature around 3.4 µm observed from ground and in situ measurements


Spectra 85

Fig. 4.7 High resolution infrared spectroscopic observations of the ν3 band (2.7 µm) of

H2O in Comet Halley taken from NASA’s Kuiper Airborne Observatory on December 24,1985. The lines originating from Ortho (O) and Para (P) H2O are resolved completely.

The spectra of the Moon is also shown for comparison (Mumma, M.J., Weaver, H.A.,Larson, H.P., Davis, D.S. and Williams, M. 1986. Science 232, 1523).

of Comet Halley and subsequently in other comets, is attributed to C−H

stretch vibrations of organics (Fig. 4.9). Most of the observed features are

attributed to methanol (CH3OH).

Since the molecule CO2 has no allowed radio transitions, it does not

fluoresce in the UV or optical regions. It can only be observed in the

infrared region. Although ν3 band of CO2 at 4.25 µm is very strong, but

cannot be observed from the ground because of strong absorption from

terrestrial CO2. It has been detected by Vega spacecraft in Comet Halley

and later on in other comets with ISO. The infrared lines of CO at 4.7 µm

arising out of V(1, 0) IR band was clearly detected in Comet Hyakutake

(C/1996 B2).

Polycyclic Aromatic Hydrocarbon molecules (PAHs) are commonly seen

from interstellar medium near 3.28, 7.6 and 11.9 µm. Some of these features

have been seen in Comet 9P/Tempel 1 from the observations carried out

with Spitzer Space Observatory (Sec. 9.5.2).



Fig. 4.8 High dispersion spectra in the 3 µm region of Comet C/1999 H1 (Lee) which

shows the lines of various hydrocarbons. The spectra was obtained on August 21, 1999.(Adapted from Mumma, M.J. et al. 2001. Ap. J., 546, 1183, Reproduced by permission

of the AAS).

The infrared observations in the region 1 to 30 µm of many comets

have shown the presence of two strong broad emission feataures around 10

and 20 µm. These features are generally attributed to silicate materials

(Sec. 9.5).

Most of the polyatomic and complex molecules have rotational transi-

tions whose wavelengths lie in the millimetre and centimetre region, i.e.

radio frequency regions. Studies based on interstellar matter showed that

interstellar clouds are dominated by molecules of simple, complex as well

as of biological importance. It was immediately felt that many of these

molecules may also be present in the cometary material. If found, it will


Spectra 87

Fig. 4.9 The infrared spectrum of Comet Halley taken on March 31, 1986 (r = 1.17 AU)

shows the presence of a broad emission feature around 3.4 µm. The dashed line refers to

the continuum for a black body of 350 K and the scattered sunlight (Wickramasinghe,D.T. and Allen, D.A. 1986. Nature 323, 44).

have a great implication with regard to the origin of comets. They could also

possibly be the parent molecules (Chap. 6) of cometary radicals. Therefore

searches for many of these molecules have been carried out on many comets

in the frequency range 80−460 GHz from ground based radio telescopes.

Large number of molecules have been identified. A typical observed spectra

from Comet Hale−Bopp (C/1995O1) is shown in Fig. 4.10 which exhibit

lines of CH3OH, SO and HC3N.

Glycine (NH2CH2COOH), an amino acid has been detected for the first

time in the dust particles of Comet Wild 2.

With regard to the spectra of diatomic molecules, the Λ−doubling of

OH around 18 cm was first detected in Comet Kohoutek in 1974. Since

then these lines have been monitored regularly in almost all the comets.

The possible presence of noble gases (He, Ne, Ar, Kr and Xe) whose

lines lie in the far UV spectral region (λ<∼ 1200 A) has been looked for, but

not detected. The detection of noble gases which are highly volatile and

chemically inactive should provide clues with regard to the thermal history

of the nucleus of a comet.



Fig. 4.10 Radio spectrum of Comet Hale-Bopp Showing the lines of molecules CH3OH

(twelve J3 - J2 A lines), SO (56-45 line) and HC3N(J(28-27)line) (Lis, D.C. et al. 1999.

Earth, Moon and Planets, 78, 13).

The studies carried out on several comets has shown the presence of var-

ious kinds of molecules of varying complexity (∼ 50), but smaller than those

detected from the interstellar medium (∼150). Many of these molecules are

likely to be parent molecules sublimated from the cometary ices. Many of

the molecules present in the interstellar medium but not seen from comets

does not mean that they are not present in comets. This could be due to

the fact that the detection of a line depend upon various factors, such as

column density of the specie, timing of the observation, possible destruction

etc.

There are large number of unindentified features that are present in

the spectra of comets in the UV, visible and radio regions. Many of these

unindentified features could be due to some already known species (atoms,

radicals, ions, simple molecules, complex molecules). Therefore the identi-

fication of these features require further laboratory studies as well as theo-

retical investigations.

The detection of continuum emission at 3.71 cm from Comet Kohoutek

in 1974 has not been detected in other comets even with more sensitive

instruments. However continuum at millimetre wavelengths have been de-

tected from Comets Halley, Hyakutake and Hale−Bopp.

The in situ mass spectrometer studies of Comet Halley has given lot of

new information about the species present in the coma. They have given

rise to the identification of a large number of new species, which is difficult

to detect through traditional spectroscopic means.

A compilation of some of the observed atomic and molecular species


Spectra 89

in comets is given in Table 4.2. It should, however, be pointed out that

the cometary coma must contain many more molecules than observed so

far. The following interesting points may be noted regarding species of

Table 4.2.

Table 4.2 Some of the observed species in comets.

Neutrals : H, C, O, S, Na, K, Ca, V, Cr, Mn, Fe, Co, Ni, Cu

C2, CH, CN, CO, CS, OH, NH, S2, SO, H2

C3, NH2, H2O, HCN, H2S, CO2, HNC, OCS, SO2, CS2

H2CO, NH3, HNCO, H2CS, CH4, C2H2, C2H6

HCOOH, HC3N, CH3CN, CH3OH,

CH3CHO, NH2CHO, HCOOCH3, HOCH2CH2OH,NH2CH2COOH

Ions: C+, CO+, CH+, CN+, H2O+, HCO+, H3O+, CO+2

Dust: Silicate

CHON

1. H2 has been detected.

2. PAHs have been detected.

3. The very first emission lines emitted by OH after the dissociation of

H2O called Prompt Emission lines have been seen.

4. The molecules detected are composed of the most abundant elements

in the universe, namely H,C, O and N.

5. Most of the elements detected comes from sun-grazing comets, presum-

ably arising from the vapourization of refractory grains.

6. Most of the species detected are organic, indicating the importance of

carbon, similar to the case of interstellar molecules.

7. Highly Complex organic molecules belonging to homologous series,

Alkanes, Alcohols, Aldehydes, Carboxylic acids and Cyanopolyynes

have been detected.

8. NH, NH2 and NH3 have all been seen.

9. Hydrocarbon such as CH4, C2H2 and C2H6 are present.

10. One important element missing from the list is nitrogen. This is due to

the fact that the resonance transition of nitrogen, which is at 1200 A,



is very close to the strong hydrogen Lyman α line. Hence, it is very

hard to detect this line. But nitrogen can be inferred from the strong

CN emission lines.

11. N2 has not been detected.

12. Most of the observed species are radicals or ions which are physically

stable, but chemically highly reactive. This means that they cannot

exist as such in the nucleus. So the general belief is that these radicals

are the by−products of the break−up of some complex molecules. This

has led to what is generally termed as ‘parent−daughter’ hypothesis

for the observed species. Several possible parent molecules have been

detected.

13. The predominance of radicals implies low densities in the coma which

in turn means that the collisions may not be very important. However,

very close to the nucleus they have to be considered.

14. Most of the products of water photodissociation have been observed.

15. There is a wide variation in the chemical composition of comets.

16. Ethylene glycol (HOCH2CH2OH) is the most abundant organic

molecule inspite of its complexity.

17. Detection of amino acid Glycine (NH2CH2COOH) in cometory dust

indicate that building blocks of life are prevalent in space.

18. Grains mostly made up of C,H,N and O called ‘CHON’ particles are

present in comets. These grains contain highly complex molecules.

19. The observed anomalous isotopic ratio of elements indicate the presence

of presolar grains.

20. Silicate dust mineralogy is highly complex.

21. Dust contain both high and low temperature condensates indicating

that large scale mixing of material must have taken place in the solar

nebula.

4.2. Forbidden Transitions

So far we have been discussing the spectra arising out of the allowed

dipole transitions which have a mean lifetime of ∼ 10−8 sec.

Many forbidden lines arising out of several atoms have been seen from

various types of astronomical objects, including comets. The mean lifetime

for magnetic dipole and electric quadrupole transitions are ∼ 10−3 and 1

sec respectively.

The well−known auroral red lines of 6300.23 A and 6363.87 A, the green


Spectra 91

line of 5577.35 A of neutral oxygen atom have been seen in many comets.

The red lines originate from the 1D upper level while the green lines arise

from the 1S level as can be seen from the energy level diagram of neutral

oxygen atom (Fig. 4.11). In the beginning there was some confusion as to

Fig. 4.11 Energy level diagram of oxygen atom showing various transitions.

whether these lines were due to the Earth’s atmosphere or were intrinsic to

the comet. This was finally resolved based on high dispersion spectra. The

red lines observed in the spectra of a comet should be Doppler shifted in a

manner similar to other cometary molecular emissions, if they are intrinsic

to the comet, while that of atmospheric origin should not be shifted. Based

on such arguments it was conclusively shown that [OI] lines are of cometary

origin. In fact the observations carried out with Fabry−Perot instrument

can separate the lines arising out of the earth’s atmosphere and from the

comet quite well. The life time of the 1D state is about 130 sec and that of1S is less than 1 sec. The red doubled at 6300.34 and 6363.77 A (1D −3P)

and the green line at 5577.34 A (1S−1D) arise out of ‘prompt’ emission,

as the photodissociation of H2O lead directly oxygen atoms in 1D and 1S

states (Sec. 3.11). Oxygen atoms that are excited to the 1S state decays to



3P state only 5% of the time and rest 95% to 1D state. The line 2972 A has

also been detected. The intensity of the red doublet is quite strong as can

be seen from Fig. 4.1. The green line is weak compared to the red line. The

green line is also blended with the cometary C2 Swan band. Even so, the

green line has been detected. Since red doublet lines are strong in comets,

they have been observed regularly and studied extensively in comets.

As in the case of oxygen, the forbidden lines of carbon arising out of1D − 3P transition at 9823 and 9849 A has also been seen. The presence

of carbon atoms in the 1D state (lifetime of 4000 sec) comes from the

observation of the resonance scattered 1P − 1D transitions at 1931 A.

4.3. Line–to–Continuum Ratio

Based on the spectra in the visual spectral region, one can classify the

observed spectra of comets into two categories: (1) strong continuum and

(2) strong molecular line emissions. Although at present a large number of

observations exist on comets taken since early times they are however not

homogeneous. This is due to the fact that the spectra have been taken with

different instruments as well as at different heliocentric distances. Also all

the available spectra are not of a very good quality. This is particularly

so for earlier spectral observations. Therefore, it is hard to make any real

meaningful ratio of continuum to line emission in comets. Nonetheless

several attempts have been made to use the available observations in order

to see any trend in the line to continuum ratios. One such study was based

on a homogeneous set of observations of about 85 comets. A statistical

study of these comets showed a wide variation in the line to continuum

ratios among these comets (Sec. 9.1.4).

It may be noted that the spectra of comets is entering a new era due

to the availability of sensitive instruments and detectors which can provide

high spatial, spectral and temporal observations. It is also possible to get

2−dimensional images. In addition, the Earth’s atmosphere is no more a

hindrance for making observations in various spectral regions due to the

use of rockets and satellites.

As can be seen from the general discussion of cometary spectra presented

so far, comets are very rich in emissions arising out of various kinds of

molecules. The next logical step is to extract the physical conditions of the

gaseous material present in the coma from a study of these lines, which will

be discussed in the following chapter.


Spectra 93

Problems

1. Describe the spectra of a Comet at far of distance from the Sun and

very close to the Sun. Why is there a difference?

2. Cometary spectra show only the emission lines. Explain.

3. Suppose one were to observe an absorption line in the spectra and the

optical depth required is 0.1. Calculate the column density of sodium

atoms for the D1 line.

4. What are the various criteria that may be used for a firm identification

of an unknown line in a cometary spectrum.

5. What is meant by forbidden line? Describe the conditions under which

forbidden lines occur. Mention some specific examples and discuss what

we can learn from them.

References

A good account of the spectra of comets can be found in the following:

1. Arpigny, C. 1965. Ann. Rev. Astr. Ap. 3, 351.

2. Swings, P. and Haser. L., 1956. Atlas of Representative Cometary

Spectra Liege: The Institute of Astrophysique.

3. Arpigny, C., Rahe. J., Donn. B., Dossin, B. and Wyckoff, S. 1997.

Atlas of Cometary Spectra, Kluwer, Dordrecht, Netherlands.

4. Cochran, A.L. and Cochran, W.D. 2002. Icarus, 157, 297.

A discussion of the observations in the Ultraviolet (5), Visible (5,6), In-

frared (7,8) and Radio (8,9) region can be found in

5. Feldman, P.D., Cochran, A. and Combi, M.R., 2005. In Comets II,

eds, M.C. Festou, H.U. Keller and H.A. Weaver, University of Arizona

Press, Tucson, p.425.

6. A’Hearn, M.F. 1982. In Comets, ed. L.L. Wilkening, Univ. Arizona

Press, Tucson p. 433.

7. Hanner, M.S. and Tokunaga, A.T. 1991. In Comets in the Post Hal-

ley Era. eds. R.C. Newburn, M. Neugebauer and J. Rahe. Kluwer

Academic Publishers p. 93.

8. Bockelee−Morvan, D., Crovisier, D., Mumma, M.J., and Weaver, H.A.

2005. In Comets II, eds, M.C. Festou, H.U. Keller and H.A. Weaver,

Univ. Arizona Press, Tucson, P.391.

9. Crovisier, J. and Schloerb, F.P. 1991. In Comets in the Post Halley

Era. eds. R.C. Newburn, M. Neugebauer and J. Rahe. Kluwer Aca-



demic Publishers, p. 149.

The first observational results for Comet Kohoutek using Fabry−Perot in-

strument is given in

10. Huppler, D., Reynolds, R.J., Roesler, F.L., Scherb, F. and Tranger, J.

1975. Ap.J, 202, 276.

The detection of amino acid Glycine is reported in the following paper

11. Elsila, J.E., Glavin, D.P. and Dworkin, J.P. 2009. Meteorites and Plan-

etary Science, 44, 1323.

The reference for the study of gas to dust ratio is the following

12. A’Hearn, M.F. et al. 1995. Icarus, 118, 223.


CHAPTER 5

Spectra of Coma

In the last chapter, we discussed the general characteristics of the spec-

tra of comets. As was pointed out, the analysis of these spectral lines can

give information about the temperature, pressure and the physical pro-

cesses responsible for producing the lines, in addition to the abundances

of the species. Here we would like to discuss in some detail a few of these

aspects, with special reference to the cometary spectra.

5.1. Fluorescence Process

In order to analyse cometary spectra it is necessary to know the mecha-

nism responsible for the excitation of cometary emissions. The absorption

of solar radiation in their resonance transitions which then trickle down to

give the radiation is called the resonance fluorescence process. Whether

the mechanism of excitation is due to the collision process with atoms,

molecules or ions, or due to the resonance fluorescence process is governed

by the relative time scales for the two processes. The collision time scale is

given by

τcoll ≈1

nσv(5.1)

where n, σ and v denote the number density, the collision crosssection and

the velocity of the species respectively. Using for the collision cross-section

with neutral atoms or ions, a typical value σ ∼ 10−16 - 10−17 cm2 and

v ≈ 1 km/sec, n ≈ 105/cm3, the characteristic collision time scale turns

out to be

τcoll ≈ 107 sec . (5.2)

This time scale is much larger than the typical time scale for absorption of

solar radiation in the visible region of say C2 and CN at a distance of 1 AU,

95



which is about 10 to 100 sec. Therefore for typical cometary conditions the

absorption of solar radiation is the main excitation process. Another obser-

vation which also supports the above conclusion is that the line emissions

seen in a cometary spectra arise mostly from the ground electronic state of

the molecule and this involves a lot of energy. It can therefore be brought

about only by the absorption of the solar radiation. However, the striking

success for the process came from the work of Swings. He noticed that

the relative intensity of rotational lines of the CN (0, 0) band seemed to

have peculiar minimum and maximum intensities at certain locations. The

pattern also changed with the Sun-Comet distance, as can be seen from

Fig. 5.1, which shows a marked difference between the two spectra taken

about 10 days apart. Comparing the position of the observed lines with the

solar spectrum, it was found that the position of minimum intensity corre-

sponded to the regions of less flux in the solar radiation field as compared

to that of maximum intensity positions. This in turn was related to the

presence or absence of absorption lines called Fraunhofer lines. In other

words, there existed a definite correlation between the absorbed solar radi-

ation by the molecule and the corresponding emission intensity. Since the

comet has a variable velocity around its orbit, the frequency of the radia-

tion that the molecule in the comet absorbs depends upon this velocity due

to the Doppler shift effect. Therefore the observed intensity pattern should

change depending upon whether the Fraunhofer lines come in the way of

absorption or not. These are also consistent with the observed intensity

patterns. This effect is generally called the ‘Swings effect’.

Let us further examine the validity of the resonance fluorescence ex-

citation process from a detailed analysis of the rotational and vibrational

spectra of molecules. The calculation of intensities of lines or a synthetic

spectrum requires the knowledge of the population distribution in different

energy levels of the molecule. This is a complicated problem as it is nec-

essary to simultaneously consider the different electronic, vibrational and

rotational levels of the molecule in the level population calculation. The

nature of the energy levels depends upon the type and structure of the

molecule. Just as an example to show the complexity of the energy level

diagram of the molecules, Fig. 5.2 shows the various energy levels and the

transitions that have to be considered for the OH molecule. Therefore it

is rather difficult to give a general set of equations for the calculation of

the population distribution of molecules. Instead, one has to consider each

molecule on a more or less individual basis depending on its energy level

structure. However, there are some simple diatomic molecules for which the


Spectra of Coma 97

Fig. 5.1 The microdensitometer tracings in the spectral region of CN violet band of

Comet Ikeya 1963 1.(a) and (b) refer to the spectra taken on the 3rd and 13th of March1963 respectively. The variations in the band structure between the two spectra can be

seen. (Taken from Whipple, F.L. 1978. In Cosmic Dust. ed. J.A.M. McDonnell, New

York; John Wiley and Sons. p. 1).

energy levels are such that it is a good approximation to consider vibra-

tional and rotational transitions separately. Therefore it is appropriate to

outline the method for the cases of vibrational and rotational levels treated

separately. The formalism can easily be adapted to take into account the

complexities of the molecule of interest. Earlier studies indicated that the

C2 molecule appears to deviate from the general behaviour of all other

molecules. Therefore, the case of C2 molecule will be considered separately.



Fig. 5.2 The energy level diagram of OH molecule showing electronic, vibrational and

rotational structure and the various transitions between them (Schleicher, D.G. andA’Hearn, M.F., 1988. Ap.J. 331, 1058).

5.1.1. Rotational structure

Consider the case of rotational structure for a given vibrational band of

a diatomic molecule. The intensity of a line depends, among other things,

on the fraction of the species in different rotational levels. The Boltzmann’s

law for the distribution of population in different rotational levels for an

assumed temperature is not a good approximation due to the fact that the

presence of Fraunhofer lines in the solar spectrum tends to make the pop-

ulation distribution in various rotational levels irregular. In addition, the

collisions which tend to produce the Boltzmann distribution, are rather in-

frequent due to low densities present in a cometary atmosphere. Therefore,

it becomes necessary to determine the resulting population distribution

from the solution of the statistical equilibrium equations which take into

account the absorption and the emission processes. For the pure radiative


Spectra of Coma 99

case, the number of molecules in an upper rotational state Nj is determined

by the balance between the number of molecules entering and leaving this

particular state. This can be written as

dNjdt

=∑P,Q,R

NiBijρij −Nj∑P,Q,R

Aji (5.3)

where Ni is the number of molecules in the lower rotational state i, such

that i < j. The summation is over P,Q and R branches. The A′s and B′s

are the Einstein coefficients and ρ is the radiation density corresponding

to the wavelength of the transition. ρ has to be corrected for the radial

velocity (Chap. 3) of the comet through the relation

λexc = λlab

(1− v

c

)(5.4)

where v is the radial velocity of the comet with respect to the Sun which

can be calculated as the orbit is known.

The detailed energy distribution of the solar radiation field with the

Fraunhofer lines present has also to be used in the calculation of ρ. This

requires a very high dispersion scan of the solar spectrum in the region of

interest. Sometimes for simplicity and to a first approximation, ‘blocking

coefficients’ (γλ) are used, which give a measure of the degree of absorption

in a given range of wavelengths. Therefore multiplying the mean radia-

tion field with the blocking coefficients, takes into account the effect of

Fraunhofer lines in an approximate manner. However, the detailed energy

distribution should be used as far as possible.

Since the physical conditions do not change drastically within the time

of observation, one can use (dNj/dt) = 0 which implies a steady state

condition. Therefore, Eq. (5.3) reduces to∑P,Q,R

NiBijρij = Nj∑P,Q,R

Aji. (5.5)

A similar expression can be written for the transitions in the lower rota-

tional state as∑P,Q,R

NjAji +Ni+1Aroti+1 = Ni

∑P,Q,R

Bijρij +Aroti

(5.5a)

where Arot represents the transition rate for pure rotational transitions

in the lower electronic state, which should be included for a heteronuclear

molecule. The total number of equations involved depends upon the number

of rotational levels in the upper and lower states. A simultaneous solution of



these equations with the added condition that the sum total of populations

in all the levels is equal to unity, i.e.

i+j∑t=1

Nt = 1 (5.6)

gives the relative populations in various rotational levels. For illustrative

purposes, Fig. 5.3 shows the relative populations of the rotational levels

of (0, 0) band of B2Σ+−X2Σ+ transition of the CN molecule for various

heliocentric distances. It shows clearly that with decrease in distance lot

more rotational levels are populated.

Fig. 5.3 The figure gives the relative population distribution of rotational levels for theground state of CN in comets for different heliocentric distances. (Arpigny, C. 1964.Ann. d’Ap. 27, 393).

In general the excitation by solar radiation of pure rotational levels

is negligible due to weakness of solar flux at the wavelength of rotational

transitions. However at large heliocentric distances the rotational excitation

from the 2.7 K cosmic background radiation may become important and

has to be considered in the fluorescence calculations.


Spectra of Coma 101

5.1.2. Vibrational structure

Let us now consider the vibrational bands of diatomic molecules. The

relative populations in different vibrational levels have to be obtained from

the solution of the statistical equilibrium equations written for each vi-

brational level. Since many electronic transitions can arise from the same

ground state, it is necessary to take into account all these electronic transi-

tions in addition to the vibrational transitions. The statistical equilibrium

of any vibrational level is determined as before, by a balance between the

transitions into and out of that particular level. The equations for the lower

and upper electronic states i and j respectively for a homonuclear molecule

can be written as follows. For the vibrational levels of the lower electronic

state

Ni

p∑k=m

Bikρik =

p∑k=m

Nk(Bkiρki +Aki), i = 1, 2, · · · (m− 1) (5.7)

and for the vibrational levels of the upper electronic state

Nj

m−1∑l=1

(Bjlρjl +Ajl) =m−1∑l=1

NlBljρlj , j = m, p. (5.8)

Here N is the fraction of molecules in different vibrational levels, such that

Σpt=1Nt = 1. The A′s and B′s are the Einstein coefficients, ρ is the energy

density of the solar radiation.

For heteronuclear molecules, the vibrational transitions within the

ground electronic state of the molecule are permitted. This term has to

be included in the above equations. For upper electronic states, these tran-

sitions are not important as the probability for electronic transitions is

much greater than that of vibrational transitions. The Einstein A value for

the vibrational transitions v → v′ is given by the expression

Avibvv′ =

64π4ν3

3hc3|Rvv′ |2. (5.9)

Here |Rvv′ |2 represent the vibrational matrix elements. These could be

calculated from quantum mechanics provided the variation of the dipole

moment function is known. However for most of the molecules the varia-

tion of the dipole moment function is not known. For such cases, one can

approximate |Rvv′ |2 as

|Rvv′ |2 = µ21|R′vv′ |2 (5.10)



where µ1 is the coefficient of the linear term in the dipole moment expansion

and

R′v+p,v =

[r2eB

ωexp−1e

(v + p)!

p2v!

]1/2

(5.11)

where p = v′ − v and the other parameters are the usual spectroscopic

constants of the molecule. This expression is a good approximation if (v +

v′)xe <∼ 1. This condition is generally satisfied for most of the molecules of

astrophysical interest.

5.1.3. Comparison with observations

The intensities or the synthetic profile of the bands can easily be calcu-

lated from the knowledge of the population distribution in different rota-

tional or vibrational levels. The calculated profiles can then be convolved

with the Gaussian profile of instrumental resolution corresponding to the

observations. The results of such calculations for the rotational profile of

the (0, 0) band of B2Σ+− X2Σ+ transition of the CN molecule for various

cases is compared with the observed profile for Comet Mrkos in Fig. 5.4.

They clearly indicate that the observed profile agrees well with the expected

profile when the effects of Fraunhofer lines are taken into account.

A more detailed and sophisticated synthetic spectrum of the same band

has been calculated. The calculated profile agrees very well with the ob-

served profile of several comets (Fig. 5.5). Such comparisons for the (0,

0) band or bands arising from A1Π− X1Σ+ transitions for several other

molecules such as CO, CS and OH have also been carried out for various

comets. They also show good agreement.

In a similar way, the expected intensities of various vibrational bands

of the molecules CO, CO+, CS, CN, CN+ etc. can be calculated based

on the resonance fluorescence process. As a typical case, Table 5.1 shows

a comparison of the expected and the observed intensities of lines for the

molecules CO+ and CS. The results for several other molecules also show

a similar agreement.

It is interesting to note that the total excitation rate to A1Π state of

CO by the absorption of solar radiation in the 1500 A region leading to

resonance fluorescence in the CO(A−X) system is around 1 to 2 × 10−6s−1

at 1 AU. This is much smaller than the excitation rate by solar radiation


Spectra of Coma 103

Fig. 5.4 Comparison of the expected and the observed (solid curve) intensity distri-

bution in the R-branch of the violet (0, 0) band of CN system for Comet Mrkos. Thecontinuous curve refers to results based on Boltzmann distribution for 500K. The dotted

and dashed curves refer to results based on statistical equilibrium calculations without

and with Fraunhofer lines taken into account, (Arpigny, C 1964, loc. cit.).

of the CO v(1, 0) band at 4.7 µm, which is around 2.6 × 10−4s−1 at

1 AU. This indicates that the populations of the ground state rotational

levels are controlled mainly by vibrational excitation with solar radiation.

The extension of cometary spectra into the ultraviolet region has made it

possible to look for some of the transitions which lie in the ultraviolet region.

In some cases there could be some disagreement between the calculated and

the observed intensities. This could arise due to the fact that there may be

disagreement in the observations and also there could be uncertainties in



Fig. 5.5 Comparison of the observed R-branch lines of (0,0) band of (B2Σ+−X2Σ+)

system of CN from Comet Austin with the calculated spectrum based on resonance

fluorescence process. The calculated spectrum is shifted by 0.25 A for easier comparison(Wyckoff et al. 2000. Ap.J., 535, 991).

Table 5.1 (A−X) Bands of CO+ and CS.

Band (CO+) Computed∗ Observeda Band (CS) Computed+ Observed

(0,0) 0.11 0.12 (0,0) 1.0 1.0b 1.0c

(1,0) 0.63 0.60 (1,0) 0.03 0.05(2,0) 1.00 1.00 (0,1) 0.13 0.12 0.11

(3,0) 1.20 1.16 (1,1) 0.07 0.08(4,0) 0.67 0.64 (0,2) 0.01 0.01

(5,0) 0.28 0.40 (1,2) 0.03 0.012

a. Comet Humason (1961e) (Arpigny, C. 1964. Ann.d’Ap.27, 406)b. Comet West (Smith, A.M., Stecher, T.P. and Casswell, L. 1980. Ap. J. 242, 402)

c. Comet Bradfield (Feldman, P.D. et al. 1980, Nature 286, 132)

*(Krishna Swamy, K.S. 1979, Ap. J. 227, 1082; Magnani, L. and A’Hearn, M.F.1986, Ap. J. 302, 477).

+(Krishna Swamy, K.S. 1981. Astr. Ap. 97, 110; Sanzovo, G.C., Singh, P.D. andHuebner, W.F. 1993. AJ. 106, 1237).

the oscillator strengths, variation of electronic transition moment with the

r-centroid and other parameter which enter into the model calculation.

For those molecules for which the oscillator strengths are not available,

it may be possible to reverse the problem and get an estimate for the

oscillator strength of the transitions from a comparison of the expected


Spectra of Coma 105

and the observed intensities of lines.

From the discussion so far, it is clear that the resonance fluorescence

process is very successful in explaining the observed band spectra of various

molecules in comets. So far, only the effect of the radial component of the

velocity of the comet in its orbit leading to Doppler shift of lines has been

considered. This is the well known Swings-effect. In addition, it is necessary

to consider the velocity of the gas in the coma. As the gas comes out of

the nucleus, it expands outwards in all directions. Therefore, the resultant

Doppler shift of the solar radiation depends upon these two components.

The resultant velocity will be different at different locations in the comet

and hence the relative intensities of the same feature should vary with

position in the coma. Such an effect was first noticed by Greenstein in the

Comet Mrkos. This is generally called Greenstein effect. This can be seen

clearly from Fig. 5.6 where the relative intensities of CN(0, 0) lines at three

different locations in the Comet Seki are shown. To take care of the effect

of internal motions in the solution of statistical equilibrium equations, it is

necessary to know the distribution of velocities of the species as a function

of the distance from the nucleus. This has to be derived on the basis of a

simple physical model. One such model could be that the gaseous material

is flowing out of the nucleus with a uniform velocity. The model could be

made more complicated by including the resistance force due to the solar

wind and so on. In addition to the orderly motion discussed so far, there

may also be motions present purely of random character. It is even harder to

consider this type of velocity in the model calculation. Therefore the subject

of the various kinds of gaseous motions in comets is quite complicated and it

is hard to disentangle one from the other. In principle, if one has a very high

dispersion spectrum, it is possible to construct a detailed model including

the effect of all the velocities, which can reproduce the observations well.

There are also other complications that have to be taken into account.

In the above treatment it is assumed that the coma is optically thin. For

reasonable value for the absorption cross section, σ ≈ 10−17 − 10−18 cm2

for molecules, the optical depth effect (τ = Nσ>∼1) has to be considered for

the molecules with a large column density N , which imply large production

rates of the molecules. Since for most of the cometary species the optical

depth τ <∼ 1, the optically thin assumption is a very good approximation

for most of the lines. But for some resonance lines such as Lyman α line

of hydrogen at 1216 A and the rotational lines of the H2O molecule, the

optical depth could be significant and therefore could affect the molecu-

lar excitation through radiative trapping. Another complication arises for



Fig. 5.6 Top section shows the rotational lines of (0.0) band of the violet CN systemin Comet Seki-Lines. The lower portion represents the tracings of the spectra in theregions a, b and c. (Arpigny, C. 1965. Mem. Acad. Roy. Belg. Cl 8, Vol. 35, No. 5).

molecules with short lifetimes. It is possible that such molecules may decay

even before the fluorescence equilibrium is reached, which require several

cycles of excitation and de-excitation. These have to be treated based on

time dependent calculations.

In addition to the radiative process collisional excitation and de-


Spectra of Coma 107

excitation could also be important. Since the density to a first approxi-

mation varies with distance from the nucleus as r−2, the collisions can be

an important excitation process close to the nucleus, ∼ a few thousand kilo-

meters for molecules with production rates ∼ 1029/sec. In the inner coma,

the molecules will be more in thermodynamic equilibrium rather than flu-

orescence equilibirum. As one moves outwards there will be a region where

both thermodynamic and fluorescence equilibria will be operating. Finally

at large distances, fluorescence equilibrium takes over. Since the observed

intensities or profile of emission lines are an average over the line of sight

in the coma, it is necessary to integrate over the whole volume, including

both collision and radiative processes. For collision effects, one generally

considers the collision of the molecule with H2O as it is the most abundant

molecule in the coma. However, other species might have to be considered

if they turn out to be important. For example, the production rate of CO

is much larger than that of H2O in comets at large heliocentric distances

and therefore collisional excitation is due to that of CO.

The collision de-excitation rate is given by

Ccoll = nH2Oσcollv (5.12)

where σcoll is the collision cross section for rotational transitions of the

molecules produced by collisions with H2O, nH2O is the number density of

H2O molecules and v is the mean relative velocity between H2O and the

molecule and is given by

v =8kT

π

[1

mH2O+

1

mmol

]. (5.13)

Here T is the temperature of the gas, mH2O and mmol are the masses of

the two molecules. The excitation cross sections can be obtained from the

de-excitation crosssections from the detailed balance condition

σ(l→ u) = σ(u→ l)gugl

exp

[−Eu − El

kTkin

](5.14)

where u and l refer to upper and lower levels and Tkin the temperature that

characterises the colliding particles.

The modelling of the collision process in the coma is rather uncertain

due to uncertainties in the collision cross sections between the molecular

species and the H2O molecule and the temperature of the coma gas. How-

ever the best available data is generally made use of in such calculations.

The variation of the H2O density with distance from the nucleus is also

required and is generally calculated based on Haser’s model (Sec. 6.1.2).



The time evolution of the profiles, until it attains the equilibrium state, can

be investigated based on the time dependent population distribution. As

an example, the results for the case of A−X(0, 0) band of CS is shown in

Fig. 5.7 for different initial population distributions. Such calculations are

important when comparing with observations made closer to the nucleus.

Fig. 5.7 Time evolution of the synthetic profile of P, Q and R branches of the (0, 0) band

of CS molecule for the case when the initial population (t = 0) is distributed over therotational levels in the lower level J ′′=10 to 19 (a) or upper levels J ′= 10 to 19 (b) at 4000

km from the nucleus. The time for each curve is marked. The curves are for a spectralresolution of 0.8 A corresponding to Halley’s data (r=1.1 au, Q (H2O)=3.0×1029 s−1,

T = 300 K; Krishna Swamy, K.S. and Tarafdar, S.P. 1993. Astr. Ap. 271, 326).

5.1.4. Case of C2 molecule

One exception to the general behaviour of the resonance fluorescence

excitation process appeared to be the case of the C2 molecule. This was no-

ticed as early as 1953 and showed that the calculated relative band strengths

did not agree with the observed relative strengths. There was a discrepancy

of nearly a factor of two between the expected and the observed intensi-

ties. Early investigators had expressed this discrepancy in terms of the


Spectra of Coma 109

vibrational temperature of the C2 molecule. Therefore it is also called ‘the

problem of low vibrational temperature of C2’. The observed difference was

also found to be dependent upon the heliocentric distance of the comet.

The excitation mechanism which populates the upper energy states also

determine the vibrational level population in the lowest electronic state.

The initial population distribution produced at the time of formation of

the molecule is masked by the excitation and decays, finally leading to the

equilibrium population distribution. The resulting population distribution

in the bottom triplet state should therefore be close to the Boltzmann distri-

bution corresponding to the colour temperature of the Sun in the Swan band

region. However the observed distribution gives a lower value ∼ 3500 K.

This discrepancy was rather disturbing in view of the fact that the C2

Swan bands are the strongest in the visual region of a cometary spectrum

and this had questioned the basic fluorescence excitation mechanism itself.

The problem of low vibrational temperature of C2 has finally been resolved

based on a new physical effect which seems to operate very effectively in the

case of the C2 molecule as can be understood from the energy level diagram

of the C2 molecule shown in Fig. 5.8. The Swan bands which are strong

in the cometary spectra arise from the triplet state, which lies about 714

cm−1 higher than the ground singlet state. This had given rise to another

interesting problem which also eluded a reasonable explanation. i.e. why

is it that Swan bands are strong in a cometary spectrum even though they

do not arise from the ground state of the molecule, contrary to the normal

situation. It was therefore not clear whether this reflected the preferential

formation of the triplet state at the time of formation of the C2 molecule

or the fact that only the weak lines of the singlet series are expected in

the spectra. In Fig. 5.8 the vibrational levels of the electronic states a3Πu

and b3Σ−g of the triplet states are also shown. It shows that the vibrational

levels of the ground triplet electronic state, for v′′ > 4, lies at a higher en-

ergy level compared to those of the b 3Σ−g state. Therefore the transitions

from higher vibrational levels of the a 3Πu electronic state can decay to

lower lying vibrational levels through the vibrational levels of the b 3Σ−gelectronic state. The result of such transitions is to depopulate the higher

vibrational levels of the a 3Πu state giving rise to more concentration in the

lower vibrational levels which implies a non-Boltzmann type of population

distribution (Fig. 5.9). This effect is in a sense equivalent in principle to

vibrational transitions in the ground state of the molecule. Therefore even

though C2 is a homonuclear molecule wherein transitions between the low-

est lying vibrational levels are forbidden, there is a physical mechanism by



Fig. 5.8 Energy level diagram for the C2 molecule. The electronic structure is shown on

the left-hand side. The singlet electronic state is the ground state of the molecule. Var-ious transitions are shown. B−R indicates the Ballik−Ramsay bands. The vibrationallevels of three lowest electronic states are shown on the right-hand side.

which the population from higher levels cascade radiatively to lower vibra-

tional levels in the ground electronic state of the molecule. This increases

the concentration in the lower levels thereby simulating a lower excitation

temperature. Since the rate of spontaneous downward transitions would

be fixed, this process would become relatively more important in the lower

radiation field at greater heliocentric distances, so that the vibrational tem-


Spectra of Coma 111

Fig. 5.9 The expected vibrational population distribution in the a3Πu state is shownfor the cases when Fox-Herzberg, Ballik-Ramsay and Swan bands (referred as All bands)

or only Swan bands are included in the calculations. (Krishna Swamy, K.S. and O’Dell,

C.R. 1977. Ap. J., 216, 158). The deviation occurs due to new depopulation mechanismdiscussed in the text.

perature would be expected to drop with increasing heliocentric distance.

In order to calculate the expected intensities of Swan and Phillips sys-

tems, the modelling of the C2 molecule has to be carried out taking into

account simultaneously both the singlet and triplet states. The model de-

veloped includes a total of ten electronic states with several vibrational

levels in each of the electronic state. The forbidden singlet−triplet ground

state transitions is the link between the singlet and triplet states for which

the electronic transition moment is taken as a variable parameter.

The solution of the statistical equilibrium equations of such a system

gives the population distribution in various levels. The resulting population

distribution in the ground state as shown in Fig. 5.9 clearly shows the

importance of the depopulation mechanism discussed earlier.

The expected band sequence flux ratios can be calculated from a knowl-

edge of the steady state relative populations in different vibrational lev-

els. The total energy emitted in a band sequence for a given value of



∆v = (v′ − v′′) can be written as

F (∆v) ≈∑V ′′

nV ′′∑∆v

BV ′′v′ρV ′′v′Pv′v′′vv′,v′′. (5.15)

Here nV ′′ is the level population and Pv′v′′ is the probability that when an

absorption takes place from the state v′′ to v′ state, it will come back to

state v′′. The above expression can be written as

F (∆v) ≈∑V ′′

nV ′′PR(V ′′,∆ν) (5.16)

where PR(V ′′,∆ν) denotes the production rate. The production rates can

be calculated from a knowledge of the molecular parameters and are given

in Table 5.2.

Table 5.2 Production rates for Swan band se-quences.

v′′ PR(v′′,∆v)PR(v′′=0,∆v=0)

∆v = −1 ∆v = 0 ∆v = +1

0 0.2745 1.000 0.08131 0.3167 0.6333 0.3484

2 0.3255 0.4153 0.5335

3 0.3109 0.2450 0.65094 0.2813 0.1351 0.7168

5 0.2438 0.0700 0.7449

6 0.2030 0.0368 0.74547 0.1643 0.0249 0.7328

8 0.1299 0.0265 0.7182

9 0.1004 0.0371 0.710810 0.0747 0.0523 0.7151

11 0.0531 0.0669 0.7180

12 0.0383 0.0673 0.640213 0.0277 0.0564 0.0517

This table shows that for the band sequence ∆v = 0 the contribution

comes mostly from the lowest states, while for ∆v = +1, higher states

also contribute appreciably. Figure 5.10 shows a comparison of the ex-

pected ∆v = +1 Swan band sequence flux ratios for several values of the

electronic transition moment, |Re|2 for the ground state singlet-triplet tran-

sitions, with observations of Comet Halley. The expected variation from

the model for transition moment ∼ 2.5 × 10−6 gives a good fit to the ob-

served variation. The calculated intensities can also explain the observed


Spectra of Coma 113

Fig. 5.10 A comparison of the expected and the observed Swan band flux ratios plotted

as a function of the heliocentric distance. The continuous lines are the model baseddependence shown for several values of transition moment for the lowest singlet triplet

transitions. The filled circles show the observed dependence for Comet Halley (O’Dell,

C.R., Robinson, R.R., Krishna Swamy, K.S. MeCarthy, P.J. and Spinrad, H. 1988. Ap.J. 334, 476).

flux ratios for the Mulliken system of the C2 molecule. The results of these

calculations also explain the observed result that the intensities of the Swan

bands are stronger compared to those of the Phillips bands. These inves-

tigations also show that the forbidden singlet-triplet transitions are quite

important. The wavelength of these transitions lies in the infrared spectral

region.

The molecule CO has a structure very similar to that of the C2 molecule

containing both singlet and triplet electronic states. The singlet electronic

state is the ground state of the molecule. However, the strong bands of

CO seen in a cometary spectra arise out of singlet states (A1Π−X1Σ+) in

contrast to the case of the C2 molecule which arise from the triplet states

(d3Πg - a3Πu). It is therefore of interest to know the physical reason for

the inversion in intensities of the singlet and triplet series for the case of

the C2 molecule. This arises basically due to the difference in the energy

level structures of the two molecules as can be seen below.



Consider the simple case of two levels with population n1 and n2 corre-

sponding to the lowest singlet and triplet states of the molecule with energy

separation ∆E. The resulting population in the upper and lower levels, n2

and n1 depend essentially on the rate of the absorption and emission pro-

cesses and this can be written as

n2

n1∝(Bρ

A

). (5.17)

Here A and B are the Einstein coefficients and ρ is the energy density of

the solar radiation. The ratio of the change in population for two values of

∆E is given by[Triplet

Singlet

]∆E1

/

[Triplet

Singlet

]∆E2

=

(λ1

λ2

)5F(λ1)

F(λ2). (5.18)

Here F(λ) represents the solar flux. The above equation shows a strong

dependence on the wavelength. Therefore the relative intensities of singlet-

singlet to triplet-triplet transitions of a molecule depend strongly on the

energy separation between the lowest singlet and triplet states which are

the 714cm−1 for C2 and 49000 cm−1 for CO.

Let us now consider the rotational structure of C2 molecule. The Swan

bands of the C2 system arise out of 3π0, 3π1 and 3π2 states. The relative

population in different rotational levels of each of these substates for a

given vibrational transition can be obtained from the solution of statistical

equilibrium equations of lower and upper states. For vibrational population

distribution, the values based on statistical equilibrium equations can be

used. The synthetic profile resulting out of the superposition of many lines

arising out of the various bands and corrected for the instrumental effect is

compared with the observed profile in Fig. 5.11.

5.1.5. Prompt emission lines of OH

Laboratory studies have demonstrated that the photodissociation of wa-

ter creates a fraction of OH in the highly excited rotational levels (upto J′ ≈22 of v′ = 0 and upto J′ ≈ 17 of v′ = 1) of the electronic state A2Σ+ (see

also Sec. 6.2.2). Since the lifetime of OH formed in the electronic state

A2Σ+ is much shorter than the collision time, the OH formed in the elec-

tronic state A2Σ+ quickly trickles down to ground electronic state X2Π by

spontaneous emission giving rise to emission lines in the UV region, (A−X)

bands. This type of transition lines are called prompt emission lines as the

dissociation fragment emit promptly after its production. The wavelength


Spectra of Coma 115

Fig. 5.11 Comparison of the calculated (continuous) and observed (dashed) spectra of

∆v = +1 sequence of the C2 Swan system for Comet West. (Observations are from Lam-

bert, D.L. and Danks, A.C. 1983. Ap. J. 268 428; spectral resolution (FWHM)=0.3 A).

of these emission lines comprising P − and Q doublets of the (0, 0) band

lie in the wavelength region of 3100 to 3250 A. The prompt emission lines

of OH are much weaker than the solar resonance fluorescence lines of OH

arising from the ground electronic state X2Π.

These studies are of direct relevance to comets as the photodissociation

of H2O by solar Lyman α photons in the coma should lead to OH in highly

excited rotational states of A2Σ+ giving rise to prompt emission lines of

(A−X) transitions in the UV region. For detecting these lines in a comet,

observations have to be carried out very close to the nucleus where OH

molecules will still be in the excited electronic state A2Σ+ produced after

the photodissociation of H2O.

Comet Hyakutake gave the first opportunity to look for these lines, as

the comet came close to the earth, ∆=0.11 AU, that made it possible to

make spectral observations very close to the nucleus. The observations were

carried out with Kitt Peak 4 m Mayall Telescope on 26 March 1996 when

the comet was at r = 1.02 AU. All the expected lines (as per Laboratory

studies) of P - and Q lines of OH upto rotational quantum number 22 of

the (0, 0) band are detected for the first time in the spectra covering the

wavelength region from 3100 to 3250 A. This is shown in Figs. 5.12 and 5.13.

The intensities of the lines decrease downwords from the rotational quantum



Fig. 5.12 The Prompt Emission lines of P and Q doublet of OH arising from the highlyexcited rotational levels of A2Σ state ((A − X) transition) was observed for the firsttime in Comet Hyakutake on 26 March 1996 (r = 1.02 AU and ∆ = 0.11 AU) in the

wavelength region 3100 to 3250 A. The above figures cover the wavelength region from3100 to 3200 A. Many of the lines marked as A and B in the spectra are the promptemission lines of OH. These are compared with the computed spectra (lower panels)which is summed over resonance fluorescence and prompt emission lines. (Meier, R. etal. 1998. Icarus, 136, 268; A’Hearn, M.F., Krishna Swamy, K.S. and Wellnitz, D.D.

2007. Bull. Am. Astro. Soc., 39, 507).


Spectra of Coma 117

Fig. 5.13 The Prompt Emission lines of P and Q doublet of OH arising from the highlyexcited rotational levels of A2Σ state ((A − X) transition) was observed for the first

time in Comet Hyakutake on 26 March 1996 (r = 1.02 AU and ∆ = 0.11 AU) in the

wavelength region 3100 to 3250 A. The above figures cover the wavelength region from3200 to 3250 A. Many of the lines marked B in the spectra are the prompt emission lines

of OH. These are compared with the computed spectra (lower panels) which is summedover resonance fluorescence and prompt emission lines. The expected intensities of P, Q

and R lines are also shown. (Meier, R. et al. 1998, Icarus, 136, 268; A’Hearn, M.F.,

Krishna Swamy, K.S. and Wellnitz, D.D. 2007. Bull. Am. Astro. Soc., 39, 507).



number 22 in conformity with laboratory studies. The observed spectra

is compared with the computed spectra which is summed over resonance

fluorescence and prompt emission lines. The two are in good agreement.

The weak features present alongwith P- and Q branches of (0, 0) band are

due to prompt emission lines arising out of (1, 1) transitions. The strong

features present in the calculated spectra are due to resonance fluorescence

process. The expected intensity distribution of all the P,Q and R branch

prompt emission lines are also shown in Fig. 5.13.

The importance of the detection of prompt emission lines of OH is that

it can help in a better understanding of the cometary coma very close

to the nucleus. In addition it can also help in the understanding of the

photodissociation of H2O process.

5.1.6. Molecules other than diatomic

The analysis of polyatomic molecules is more complicated than that of

diatomic molecules. Therefore, their studies are limited in extent. The

work done on molecules like H2O, NH2, CO+2 , NH3, etc., indicate that the

resonance fluorescence process is the major excitation mechanism for these

molecules as well.

5.1.7. OH radio lines

The hydroxyl radical gives rise to lines in the radio region due to Λ-

splitting of the levels (Table 3.1). The line at 1667 MHz which occurs at

a wavelength of 18 cm was first detected in Comet Kohoutek in 1974. The

most studied lines in comets are the 1665 and 1667 MHz lines. The inten-

sity of the lines are found to be variable. Curiously, one finds that these

lines have been seen in the emission as well as in the absorption which

depend on the heliocentric distance of the comet. These observations can

be explained by the fluorescence excitation process. Since the transitions

between the Λ-doublet levels are highly forbidden, in the absence of any

other excitation mechanism, the levels should be populated according to

their statistical weights at the coma temperature. The deviation from this

equilibrium value is generally referred to as inversion or anti-inversion. The

cosmic microwave background radiation of 2.7 K and the galactic sources

can induce transitions. The line may be seen in absorption if the population

is anti-inverted or in emission if the population is inverted due to stimulated

emission. The Λ-doublet levels are actually populated through the process


Spectra of Coma 119

of UV solar absorption lines from the ground state 2Π3/2 (v′′ = 0, J = 3/2)

to the state 2Σ+, v = 0 and v = 1 levels and then cascade back to the

ground level. Since the UV absorption lines or the UV fluorescence efficien-

cies depend upon the comet heliocentric velocity (Swings effect, Fig. 5.14),

the resulting relative population distributions in the four levels also de-

pends critically on the heliocentric velocity of the comet and hence on the

heliocentric distance. This can result in the lines of OH to be produced

Fig. 5.14 The g-factor or the Fluorescence efficiency factor for the (0,0) band of OH is

shown as a function of the heliocentric radial velocity. The variation in the value canclearly be seen arising due to Swings effect (Schleicher, D.G. and A’Hearn, M.F., 1988.

Ap.J. 331, 1058).

either in absorption or in emission. Such an effect can clearly be seen from

the observation of Comet Kohoutek which is shown in Fig. 5.15. The

upper section of the curve shows the observed peak antenna temperature

of 1667 MHz line as a function of time. The lower section shows the ex-

pected inversion based on the ultraviolet pumping model and is given by

[(nu − nl)/(nu + nl)]. Here nu and nl represent the projected densities of

the upper and lower levels of the Λ−doublet. There is a one-to-one rela-

tion between the two curves. This mechanism has been verified for a large

number of radial velocities as well as observations based on several comets.

This explanation is generally accepted and is used in interpreting the 18 cm



Fig. 5.15 The radio OH data for Comet Kohoutek. The top portion denotes the ob-

served peak antenna temperature of the 1667 MHz line as a function of time. The lower

portion denotes population inversion predicted by the ultraviolet pumping model as afunction of time (Adapted from Biraud, F. Bourgois, G., Crovisier, J. Fillif, R., Gerard,

E. and Kazes, I, 1974. Astr. Ap. 34, 163).

observations. Therefore, the ultraviolet pumping of the OH radio lines is

the dominant excitation mechanism. However, the collisions can also be

quite effective as the Λ−doublet transitions are forbidden and the radia-

tion field is weak. It is found that collisions with ions and electrons can

be more important than even neutrals. This could have a severe effect in

influencing the inversion in the inner coma of comets with high production

rates. This is recognized as an important process in the calculation of radio

emission from comets.


Spectra of Coma 121

5.1.8. Oxygen lines

Allowed transitions. The triplet lines of oxygen at λ=1304 A (1302.17 A)

are very strong in the UV region of the cometary spectra (Fig. 4.4). Of

particular interest are the Comets Kohoutek and West in which these lines

have been seen and these comets had heliocentric velocities in the range

of about 44 to 55 km/sec. To see whether the excitation process of these

lines is through resonance fluorescence process, a high dispersion spectrum

of the solar line at 1302.2 A is shown in Fig. 5.16. The line width of

±40 km/sec corresponds to a Doppler shift of about ±0.17 A. For r > 40

km/sec, the cometary absorption wavelength will be completely shifted

from the solar line. Therefore, for comets with heliocentric velocity r <∼ 30

km/sec, the emission is produced by the resonance scattering of the solar

lines. For higher velocities, either the line should be very weak which is not

the case or there should be some other excitation mechanism. From the

energy level diagram of the oxygen atom (Fig. 4.11) it can be seen that

the line 1025.77 A is very close to the solar Lyman β line of hydrogen of

1025.72 A. Therefore excitation is produced mainly through chance coin-

cidence with solar HI Lyman β and hence absorption takes place. It then

goes through cascading process through intermediate levels leading finally

to the emission line at 1304 A. This mechanism is well known in planetary

nebulae as Bowen mechanism. The intensity of OI 1304 A multiplet pro-

duced by resonance scattering of the solar lines is much stronger than those

produced by the Lyman β fluorescence. Therefore the intensity of the emis-

sion line 1304 A depends critically on the radial velocity (Swings effect),

the width of the solar emission line and the width of cometary absorption

line.

5.1.9. Forbidden transitions

The identification of the red oxygen doublet lines at 6300 and 6364 A

in comets immediately raises the question of the excitation mechanism of

these lines as they are electric dipole forbidden transitions. First of all, it

is interesting to investigate whether the fluorescence process can or cannot

explain these lines. The direct method to test this hypothesis is to com-

pare the fluorescence efficiency rate for the red oxygen lines with those of

ultraviolet resonance lines which is given in Table 5.3.



Fig. 5.16 High resolution profile 1302.2 A line of OI in the Sun (Solid curve). Thedashed curve represents the g-factor for fluorescence excited by solar Ly β as a function

of heliocentric velocity (Feldman, P.D., Opel. C.B., Meier, R.R. and Nicolas, K.R. 1976.The Study of Comets, eds. B. Donn et al. NASA SP-393, Washington, D.C. p. 773).

The values show that the excitation to 1S0 level (2972 A) is negligible

(Fig. 4.11). Also the expected intensities of the line 6300 A can be calcu-

lated based on the observed intensities of the 1304 line and the g-factors

of Table 5.3. The expected intensities of red lines are very much smaller

than those of the observed values. Therefore, the resonance fluorescence

process cannot be responsible for the excitation of forbidden oxygen lines.

The collisional excitation is also inadequate for explaining the observed

emission.


Spectra of Coma 123

Table 5.3 Fluorescence effi-

ciency factors at 1 AU for oxy-gen.

Line(A) g-factor∗

(photons/sec/atom)

989 1.6 (-8)1027 0.4 to 1.5 (-6)

1304 0.6 to 1.5 (-5)a

1.1 to 3.9 (-7)b

2972 3.1 (-15)5577 6.21(-14)

6300 4.2 (-10)

6364 1.3 (-10)

* See Chap. 6.

Number in the bracket refers topower of 10 (a) for solar ab-

sorption; (b) Lyman β induced

fluorescence; (Festou, M.C. andFeldman, P.D. 1981, Astr. Ap.

103, 154).

Therefore it is generally assumed that the photo dissociation of some

molecules leave oxygen atoms in the excited 1D state which then decay to 3P

ground state giving rise to 6300 A emission. 1D state can be generated from

the photodissociation of molecules H2O, OH, CO2 and CO (Sec. 6.2.9).

However, the contributions from the photodissociation reactions of CO2

and CO are much smaller than that of H2O.

5.1.10. Molecular band polarization

The resonance fluorescence excitation process which is responsible for

the observed emission lines in comets leads to two other important effects.

The first being that even though the incident radiation is a natural one, the

fluorescence lines should be linearly polarized. Secondly the total intensity

of emission is not isotropic but depends upon the phase angle.

The linear polarization of emission bands in comets was first observed in

Comet Cunningham 1941 I from the study of C2 and CN bands. This was

followed by observations of other comets. The polarization of C2 (5140), C3

(4060), CN (3871) and OH (3090) bands have been measured in Comets

Halley and Hartley - Good 1985 XVII. The polarization of the bands of

CO+ (4260) and H2O+ (7000) has also been measured in Comet Halley.



The measured linear polarization in percentage for Comet Halley at a phase

angle ∼ 50 is around 3.9, 4.0, 3.8 and 1.8 for the molecules CN, C2, C3

and OH respectively.

There exists several studies pertaining to the polarization of fluores-

cence lines, both from experimental and theoretical points of view. The

theoretical studies are based on the application of the principle of Zeeman

splitting in a magnetic field. Therefore, the important parameter in deter-

mining the polarization in a molecular band u → l (u is the upper level

and l is the lower level) is the Larmor frequency νL of the molecule. In the

case of a strong magnetic field, 2πνL > Au→l where Au→l is the Einstein

coefficient for the transition, the molecule will precess strongly in the up-

per state before de−excitation takes place and therefore it essentially looses

memory of its unidirectional excitation process. The resultant effect being

that the fluorescence emission line will be isotropic and unpolarized. For

the other case of a weak magnetic field, 2πνL < gl→u, where gl→u is the ex-

citation rate of the band, the fluorescence cycle will lead to alignment of the

molecule with respect to the direction of the excitation field. The typical

values for a cometary molecular band is, Au→l ∼ 106/sec and gl→u ∼ 103

to 10−1/sec. The general value of the magnetic field as measured by space-

crafts in Comet Halley is of the order of a few 10 nT. This corresponds to

νL ∼ 102/sec. These values show that

gl→u < 2πνL < Au→l.

The resultant effect being that the molecules will precess in the lower state

between the fluorescence cycles, but not in the upper state. This therefore

preserves the polarization for the case of anisotropic excitation.

The excitation by unidirectional natural radiation leading to linear po-

larization of an emission line has maximum polarization for a phase angle

of 90o (i.e. observed perpendicular to the exciting radiation). The expected

variation of polarization with phase angle θ is given by the expression

P (θ) =Pmax sin2 θ

1 + Pmax cos2 θ(5.19)

and the total intensity should vary as

Itotal =Pmax cos2 θ + 1

13Pmax + 1

. (5.20)

Here Pmax represents the maximum polarization corresponding to θ = 90.

The value of Pmax depends upon different rotational quantum number J for


Spectra of Coma 125

the P, Q and R branches of the band. Theoretical calculations are available

for some of the transitions at the present time.

The value of Pmax for various molecular bands can be determined from

the observed polarization and phase angle corresponding to the time of

observation. The derived value of Pmax from observations for the C2 and

CN molecules is in agreement with the expected theoretical value of Pmax

of 7.7%. The observed polarization variation also established the validity of

the theoretical dependence given by Eq. (5.19). The Pmax values derived

from observations for the C3 molecule ∼ 6% which is similar to that of

C2 and CN, but smaller than the expected value of 19%. During the 2003

apparition of Comet 2P/Encke the observed polarization of C2 and CN were

in good agreement with the theoretical calculations. However for C3 the

observed polarization was close to the value of CN and C2 (∼ 7%) except

on one day when the observed polarization was 16.4% which is close to the

expected value of 19%. The derived Pmax value for H2O+ and CO+ is quite

high ∼ 20%.

5.2. Excitation Temperature

The population distribution under thermal equilibrium is given by the

Boltzmann law. Even though it does not represent a real situation, it can

be used to a first approximation for estimating the excitation temperature.

This excitation temperature is found to be not much different from the

temperature obtained based on a detailed analysis of the level population

in many cases.

5.2.1. Rotational temperature

The temperature determined from the application of the Boltzmann

law to rotational lines is generally called the rotational temperature. For

rotational levels, the exponential factor e−E/kT is given by e−BJ(J+1)hc/kT

where B is the rotational constant of the molecule (Chap. 3). In addition,

for each level there are (2J + 1) states. The number of molecules in any

rotational level J in terms of the total number N can be expressed as

NJ =N

Qrot(2J + 1)e−BJ(J+1)hc/kT . (5.21)

Here Qrot is the rotational partition function and is given by

Qrot =∑

(2J + 1)e−BJ(J+1)hc/kT . (5.22)



The rotational population NJ , goes through a maximum and then decreases

with an increase in J . For a given temperature, the maximum value of J

for which the population is maximum can be evaluated from the above

equation and is given by

Jmax =

√kT

2Bhc− 1

2. (5.23)

The intensity of a rotational line in the emission can be written as (Chap. 3)

Iem = Cν4SJe−BJ(J+1)hc/kT . (5.24)

Here C is a constant and SJ is the Holn-London factor. T is the rotational

temperature. A plot of log (Iem/SJ) as a function of J(J + 1) results in a

straight line. From the slope of the line the rotational temperature can be

evaluated.

The above method has been applied to the case of cometary spectra

to obtain the rotational temperature of various molecules. One can get a

better estimate of the rotational temperature if the rotational levels are

populated to high quantum numbers. In addition one requires the spectra

of high dispersion for the lines to be resolved. Studies have been carried

out for molecules like CH, CN, OH, C2, etc. Except for the C2 molecule,

the rotational temperatures for all other molecules have values in the range

of about 200 to 400K. The rotational temperature for the C2 molecule

based on the profile analysis of well-resolved lines of Swan bands gives the

temperature values in the range of about 3000 to 4000K (Fig. 5.11). The

vast difference in the rotational temperature between the C2 and other

molecules arises due to the fact that for heteronuclear molecules like CH,

CN, etc., the transitions in the ground state of the molecule are possible;

these radiate away the energy. This makes the temperature low. But for

the homonuclear molecule C2, such types of transitions are forbidden and

hence give higher temperatures.

The (0, 0) band spectra of Swan system for Comet Halley taken at a

high resolution of 0.06–0.08 A (FWHM) resolved all the rotational lines of

the R branch. The observed intensities of these rotational lines indicated

the existence of two vastly different well defined Boltzmann temperature

population distributions corresponding to those of lower and higher rota-

tional levels. In particular, the low rotational levels with J ′ <∼ 15 gave a

rotational temperature, Trot ∼ 600 − −700 K, while the higher levels gave

a rotational temperature, Trot ∼ 3200 K. The synthetic spectra based on


Spectra of Coma 127

time-dependent rotational population distribution with resonance fluores-

cence excitation process show that a reasonably good fit over the whole

wavelength region of the observed spectra, 5162–5132 A, can be obtained

for a time interval of around 2× 103 sec as shown in Fig. 5.17. This time

Fig. 5.17 Comparison of the calculated time dependent spectra (Solid line) with the

observed spectra (dashed line) for the (0, 0) Swan band in the wavelength region 5166lto 5154 A of Comet Halley. The calculated curves are for a time interval of 2 × 103

secs and FWHM=0.06 A. The short line with filled circles show the location of the NH2

features. Several of the P, Q and R lines are also marked (Krishna Swamy, K.S. 1991.Ap. J, 373, 266).



scale is consistent with the time ∼ 2.7× 103 sec taken by the C2 molecules

with a velocity ∼ 1 km/sec, to cross the equivalent radius of the projected

area on to the sky. This indicates that the observations have picked up only

molecules with the time interval ∼ 2× 103 sec from the time of formation

indicating that the level populations does not appear to have reached the

steady state values.

5.2.2. Vibrational temperature

A mean vibrational excitation temperature may be defined from the

Boltzmann law representing the population distribution in various vibra-

tional levels. An expression for vibrational temperature can be written in

an analogous manner to that of rotational temperature as

Nv =N

Qvibe−G(v)hc/kT (5.25)

where Qvib = e−G(v)hc/kT is called the vibrational partition function. G(v)

represents the energy of the vibrational level v. The emission intensity of

a vibrational band can be written as (Chap. 3).

Iem = Cqv′v′′e−G(v)hc/kT (5.26)

where qv′v′′ is the Franck-Condon factor for the band (v′, v′′) and C is a

constant. A plot of log (Iem/qv′v′′) versus G(v) gives a straight line from

which the vibrational temperature can be determined.

Several attempts have been made to get an estimate of the vibrational

temperature of molecules. However, one difficulty with this method is that

it is hard to get many bands where the contamination from other bands or

from other lines is negligible. So one usually uses the ratio of intensities

of two bands or more bands if available. For the case of C2 molecule, the

band sequence flux ratios corresponding to ∆v = 0 and +1 is generally

used. The vibrational temperature vibrational temperature obtained for

C2 molecule is around 3000 to 5000K. This is of the same order as the

rotational temperature.

5.3. Abundances of Heavy Elements

The emission lines of heavy elements are seen only for the heliocentric

distance ' 0.15AU. Comets passing through such small perihelion distances

are very rare. One such comet was the Sun grazing Comet Ikeya-Seki in


Spectra of Coma 129

1965 which had a perihelion distance ∼ 0.005 AU. This comet provided the

opportunity for observing emission lines of various elements like Na, K, Ca,

Ca+, Fe, Ni, Mn and so on in the visible region. This made it possible to

make an abundance analysis of these elements. These elements are believed

to come from the vapourization of the refractory grains.

In a steady state, the relative populations in the upper levels are gov-

erned by the Boltzmann’s formula reduced by the dilution factor W , which

is a measure of the deviation of the energy density of radiation from the

equilibrium value i.e. uλ ∝ W λ−5 10−θχ. Therefore the intensity of an

emission line is given by

I =

(8π2e2h

m

)Ngf

uλ310−θχ (5.26a)

where χ is the excitation potential of the line and θ = (5040/T ). T is

the excitation temperature corresponding to the colour temperature of the

exciting solar radiation. The term represented by the power of 10 corre-

sponds to the exponential factor in the Boltzmann distribution function. N

is the total number of atoms/cm2 in the line of sight and u is the partition

function of the line. The other quantities have their usual meanings. The

above equation can be written as

log

(λ3I

gf

)= −θχ+ constant. (5.27)

Therefore, a plot of log (λ3I/gf) against the excitation potential χ for all

the observed lines will result in a straight line whose slope gives the value

of θ. Figure 5.18 shows one such typical plot for FeI lines. The straight

line drawn through the points is for θ = 1.0. Knowing the excitation

temperature, the total number of atoms can be calculated easily from

logN = constant + log(λ3I)− log(gf)

+ log u+ θχ

The procedure can be repeated for all the observed lines. The value of

θ = 1.0 also fitted the other lines. Therefore the number of atoms of

various kinds can be determined. The calculated relative abundances of

various elements relative to iron is similar to that of the solar value.

The in situ measurements of Comet Halley, made with impact ionization

mass spectrometer on board the Vega spacecraft, have given the elemental

compostion of dust. The contribution from the gaseous component has to

be added to the dust component, to get an estimate of the total elemental

abundances. Using a mean value for the dust to gas ratio of 0.8, the derived



Fig. 5.18 A plot of log (λ3I/gf) as a function of the excitation potential of the line.

The straight line drawn through the points is for θex = (5040/Tex) = 1.0 (Adaptedfrom Arpigny, C. 1977, Proceedings of the Robert A. Welch Foundation Conferences on

Chemical Research XXI, Cosmochemistry. Houston, p. 9; see also Preston, G.W. 1966.Ap.J. 147, 718).

total abundance of gas and dust for Comet Halley is compared with the

Sun in Fig. 5.19. It shows that the elemental abundances in Comet Halley

are very similar to the solar values except for hydrogen.

5.4. Isotopic Abundances

The study of the isotopic abundances in comets has attracted consid-

erable attention as it has significant amount of information in it as to the

conditions which prevailed at the time of formation of these objects. For-

tunately, the most abundant elements, namely, H, C, N and O do have

many stable isotopes. Therefore a comparison of the isotopic ratios of


Spectra of Coma 131

Fig. 5.19 A comparison of the elemented abundances in Comet Halley and in the Sunshowing that they are similar. The abundances refer to log N(H) = 12.0 (Delsemme,

A.H. 1991. In Comets in the Post-Halley Era, eds. R.L. Newburn, Jr. et al., Kluwer

Academic Publishers, p. 377).

these elements in different kinds of objects will reveal the history of the

whole evolutionary process. The detection of several complex molecules in

comets has given the general feeling that the cometary material and the

interstellar material could be very similar in nature. The isotopic ratios

could help in clarifying this problem as well.

For a diatomic molecule, the vibrational and rotational isotopic shifts

are proportional to(

1−√

µµi

)and

(1− µ

µi

)respectively, where µ and µi

are the reduced masses of the normal and the isotopic molecule. This shows

that the expected isotopic shifts from molecules are quite small. Therefore,

in order to study the isotopic molecule, it will be a great advantage if the

abundance of the isotopic molecule is large, the isotopic line is well sepa-

rated from the line of the normal isotope and also lie in the less crowded

region of the spectrum. This introduces severe constraints on the selection

of the molecules. In addition, the spectra should be of higher resolution

to separate the isotopic line from the normal line. These restrictions auto-

matically limit the study to bright comets. Due to the difficulties involved



in the spectral and radio astronomical observations the studies have been

limited to only a few isotopic species. In principle, the isotopic ratios can

be determined better than the absolute abundances as they are not model

dependent.

The isotopic ratio 12C/13C has been determined for several comets based

on the analysis of the Swan band intensities of the C2 molecule. It is well

suited for the study as the (1, 0) band of the Swan system which occurs

at 4737 A is well separated from the 12C/13C band which occurs around

4745 A.

A typical medium resolution scan in the region of interest for the Comet

Kohoutek is shown in Fig. 5.20. From a comparison of the observed in-

tensity ratios of these two bands, it is possible to get an estimate for the

isotopic ratio of 12C/13C. This method has been applied to many comets.

Unfortunately the isotopic line at 4745 A is strongly blended with the emis-

sion lines of NH2 molecule. The high resolution scan around 4745 A line

shows that the blending comes from 4 lines of NH2. The blending problem

can be taken care with high spectral resolution observations and with bet-

ter signal-to-noise ratio. To avoid the blending problem, the possibility of

using the (0, 0) band of the Swan system has been considered. With a high

resolution spectra of Comet West it was possible to resolve completely the

rotational structure of the (0, 0) band. However, the isotopic features are

rather weak, which makes the derived isotopic ratio uncertain.

The 3883 A (0, 0) band of B2Σ−X2Σ system of CN has also been used for

the determination of carbon isotopic ratio in comets. However, the normal

lines of 12C15N, 13C14N, weak lines of (1, 1) band and P-branch lines of

the CN system also lie in this wavelength region (Fig. 5.21). All these lines

can be resolved in a high resolution spectra. Therefore the isotopic ratio

of carbon can be determined. The results of 12C/13C studies for various

comets are summarised in Table 5.4.

The deuterium-to-hydrogen ratio is of great interest as deuterium is

most likely to be synthesized in the early universe. The neutral mass spec-

trometer on board Giotto spacecraft made measurements of the compos-

tion of neutral and ion species in Comet Halley with high mass resolution.

Therefore it is possible to get the isotopic ratio D/H. The derived ratio from

the study of HDO/H2O is around 3 × 10−4. Subsequently HDO was de-

tected in Comets Hyakutake and Hale-Bopp from observations of 101− 000

rotational line at 464.925 GHz. Following this, HDO was also detected from

211−212 transition at 241.562 GHz and 312−221 at 225.897 GHz in Comet

Hale Bopp.


Spectra of Coma 133

Fig. 5.20 Photoelectric scans of Comet Kohoutek in the region of 4737 A corresponding

to (1, 0) band of the Swan System. Resolution is 0.4 A. The 12C/13C feature is around4745 A which is blended with NH2, (Danks, A.C., Lambert, D.L. and Arpigny, C. 1974.

Ap. J. 194, 745).

The molecule DCN has been seen through their rotational transition

J(5-4) at 362.046 GHz. The derived D/H ratio is 2.3 × 10−3. The rotational

transition of HC15N, H13CN and C 34S has also been detected in Comet

Hale-Bopp leading to their isotopic ratios (Table 5.4). The line of J(3-2) of

H13CN has been detected in Comet Hyakutake.

The fundamental ortho line of H182 O at 547.7 GHz has been seen in

Comet Ikeya-Zhang. The observed ratio of H182 O/H16

2 O is in agreement

with the ratio derived from mass spectroscopy from Comet Halley. The

results of the measured isotopic ratios are summarized in Table 5.4.



Fig. 5.21 The high resolution spectrum of Comet Halley in the wavelength region from3866 to 3874 A. The R-branch lines of 12CN and 13CN overlap in this region. The

features marked a to g and i belong to the (0, 0) band of the 13CN molecule while thosemarked with h and j to m are P-branch lines from the (1, 1) band of the 12CN molecule

(Jaworski, W.A. and Tatum, J.B. 1991. Ap. J., 377, 306).

All the comets indicate D/H ratio in H2O ∼ 3 × 10−4. The cometary

material is enriched by a factor of around 10 as compared to the D/H

ratio in the primordial solar nebula (∼ 2.5× 10−5) (Fig. 5.22). It is highly

likely that this enrichment is due to fractionation effect arising through

ion-molecule reactions or grain-surface processes in the solar nebula phase.

The lower value of D/H found in planets in comparison to comets appears

to be consistent with the idea that the giant planets formed around icy

cores with high cometary D/H ratio gathered a large amount of H2 from

the solar nebula with a low protosolar D/H at a later stage.

The other measured isotopic ratios of 12C/13C, 16O/18O and 32S/34S

are consistent with the solar system value of 89, 500 and 24 respectively.


Spectra of Coma 135

Table 5.4 Observed isotopic ratios.

Ratio Molecule Comet Value

D/H H2O Halley 3.08× 10−4

Hyakutake 2.9× 10−4

Hale-Bopp 3.3× 10−4

HCN Hale-Bopp 2.3× 10−3

12C/13C C2 four Comets (a) 93± 10CN Halley 95± 12

CN five Comets (b) 90± 10

CN Hale-Bopp 165± 40CN C/2000 WM1 115± 20

HCN Hale-Bopp 111± 12

HCN Hale-Bopp 90± 15

14N/15N CN Hale-Bopp 140± 35

CN C/2000 WM1 140± 30

HCN Hale-Bopp 323± 46HCN Hale-Bopp 330± 98

16O/18O H2O Halley 470± 40

H2O 153P/2002C1 450± 50

32S/34S S+ Halley 23± 6

CS Hale-Bopp 27 ± 3H2S Hale-Bopp 17± 4

(a) Ikeya, Tago-Sato-kosaka, Kohoutek, Kobayashi-Bergar-Milon

(b) Halley, Levy, Austin, Okazaki-Levy-Rudenko, Hale-

BoppAdapted from: Bockelee-Morvan, D.2005, In Comets II,

Eds. M.C. Festau, H.U. Keller and H.A. Weaver. Univ.

Arizona Press, Tucson, p. 391.

The results for 14N/15N is rather puzzling. The high spectral resolution

of CN observations for several comets give 14N/15N ∼ 150, while radio line

observation give 14N/15N ∼ 300 which is close to the terrestrial value of 270.

The lower ratio of 150 observed in Comets Hale-Bopp and WM1(LINEAR)

point to an additional source of CN other than HCN and heavily enriched

in 15N which is still not identitifed.

The measured isotopic ratios given in Table 5.4 refer to the bulk compo-

sitions. However it is possible that departures from the mean value may be

present at the microscopic level. Such deviations to the extent of 12C/13C

ratios as high as 5000 has been found in the individual small grains in the

coma of Comet Halley. The variations in 12C/13C of around 2 to 7000 has

also been seen in primitive meteorites. The wide ranges in the observed



Fig. 5.22 The deuterikm to hydrogen ratio in comets is compared with the values

observed in planets and solar nebula (photosolar) (Attwegg, K. and Bockelee-Morvan,D. 2003. Space Sci. Rev., 106 139: Hersant, F., Gautier, D. and Hure, J.M. 2001.

Ap.J., 554, 301).

value of 12C/13C ratios indicate that these particles are preserved preso-

lar grains that have survived from physical and chemical processes in the

interstellar and early solar system environments.

The measured isotopic ratio 12C/13C from the interstellar medium at

the present time seems to vary from 20 to 80. The former and latter values

corresponding to the galactic centre region and around solar neighbourhood

respectively. This is smalller compared to the cometary value of around 90.

The equilibrium value of 12C/13C based on the hydrogen burning in stars

(i.e. CNO cycle) is around 4. The general reduction in the ratio of 12C/13C

in interstellar matter compared to the solar system value is more likely to

be produced as a result of the addition of 13C. This could be brought about

by the processing of 13C by the CNO cycle in stars. On the basis of the

present picture, the processed material of the stars find their way finally

into the interstellar medium and thus enrich the medium with 13C isotopes.

The observed isotopic ratio of 12C/13C between the solar system and the

interstellar value appears to be consistent to a first approximation with the

scenario of chemical evolution which has taken place for the last 4.6 Gyr.


Spectra of Coma 137

Lastly, mention may be made of the isotopic ratios of light elements like

Li, Be, B, etc., as these elements will be destroyed easily in stars even if

they were present in the original material. The detection in comets imply

that they ought to have been produced more recently by different physical

processes. However, the detection of light element isotopes in comets is a

difficult problem.

5.5. Summary

The analysis of the spectra shows that the resonance fluorescence pro-

cess is the excitation mechanism of the observed emission lines in comets.

The abundance of various elements obtained from the study of the spectra

of Comet Ikeya-Seki as well as in situ measurement of Comet Halley show

values very similar to those of solar abundances. In addition, the isotopic

ratio of 12C/13C in comets has a value of about 90, which is the same as

the solar system value. However 12C/13C as high as 5000 has been seen in

some of the individual dust grains in Comet Halley indicating that they are

presolar grains.

Other isotopic ratios such as 16O/18O and 32S/34S are found to be simi-

lar to the solar values except the D/H ratio. It appears that the abundance

of deuterium is enhanced compared to the solar values by a factor of 10 or

so. The likely mechanism of such an enhancement could be through the

ion-molecular reactions.

Problems

1. Consider the (B−X) transitions of the molecular hydrogen. Write

down the equation between the upper and lower populations, N(v′)

and N(v′′), assuming only the absorption and spontaneous emission

terms are important. If the population in the lower vibrational levels is

determined by the Boltzmann distribution at temperature T = 2000K,

calculate the population in the upper vibrational levels and the result-

ing emission.

2. Consider the ground electronic state (X1Σ+), the first singlet (A 1II)

and the triplet (a 3II) electronic states of the CO molecule. From the

statistical equilibrium equations, obtain an expression for the intensity

ratio of (a3II−X1Σ+) to (A1II−X1Σ +) transitions in terms of the

molecular parameters of the molecule and its numerical value. Assume



ρB21 < A21 and ρB31 < A31. This problem will give an idea of the ex-

pected intensities of forbidden transitions compared to those of allowed

transitions.

3. Calculate the wavelength of a solar photon of λ =5000 A at the comet

which is at r = 0.52 AU. The comet is moving in an elliptical orbit

with e = 0.8 and has a period of 5 years.

4. The relative intensities of the rotational lines of molecular spectra in

a comet will vary irregularly with rotational quantum number and the

intensity of an individual line will vary irregularly with the distance of

the comet from the Sun. Explain why these effects occur and describe

how in principle, you could compute the line intensities.

5. In terms of the spectral features observed in the coma of a comet, is it

possible to infer the abundances of molecules in the primordial nebula?

Explain.

References

The effect of Fraunhofer lines on cometary spectra was first pointed out in

1. Swings, P. 1941. Lick Obs. Bull. 19, 131.

The application of Resonance Fluorescence Process to molecules can be

found in

2. Arpigny, C. 1965. Mem. Acad. Roy. Belg, Cl. 8, 35, 5.

For some later work, the following papers may be referred.

3. Krishna Swamy, K.S. 1981. Astr. Ap. 97, 110.

4. Weaver, H.A. and Mumma, M.J. 1984. Ap.J. 276, 782.

5. Kleine, M., Wyckoff, S., Wehinger, P.A. and Peterson, B.A. 1994. Ap.

J. 436, 885.

6. Magnani, L. and A’Hearn. M.F. 1986. Ap.J. 302, 477.

7. Schleicher D.G. and A’Hearn. M.F. 1988. Ap.J. 331, 1058.

8. Reyle, C. and Boice, D.C. 2003. Ap. J., 587, 464.

For the calculation of vibrational transition probabilities, the following book

may be referred to:

9. Penner, S.S. 1959. Molecular Spectroscopy and Gas Emissivities, Read-

ing: Addison-Wesley Publishing Company.

The early work on C2 molecule is discussed in

10. Stockhausen, R.E. and Osterbrock, D.E. 1965. Ap.J., 141, 287.

For later work, the following papers may be referred:

11. Krishna Swamy K.S. and O’Dell. C.R. 1987. Ap. J. 317, 543.


Spectra of Coma 139

12. O’Dell, C.R., Robinson, R.R., Krishna Swamy, K.S., Spinrad, H. and

McCarthy, P.J. 1988. Ap.J. 334, 476.

13. Gredel, R., van Dishoeck, E.F. and Black, J.H. 1989. Ap. J. 338, 1047.

14. Combi, M.R. and Fink, U. 1997. Ap.J., 484, 879.

Papers pertaining to Prompt Emission lines.

The extensive Laboratory study carried out on H2O is coverred in the fol-

lowing paper

15. Carrington, T. 1964. J. Chem. Phys., 41, 2012.

The following papers cover the application to Comets.

16. Bertaux, J.L. 1986. Astron. Astrophys., 160, L7.

17. Budzien, S.A. and Feldman, P.D. 1991. Icarus, 90, 308.

Comet Hyakutake spectral observations is in the following paper:

18. Meier, R. et al. 1998. Icarus, 136, 268.

The first detection of Prompt Emission lines in Comet Hyakutake is cov-

ered in the paper

19. A’Hearn, M.F., Krishna Swamy, K.S. and Wellnitz, D.D. 2007. Bull.

Am. Astro. Soc., 39, 507.

The following papers deal with polarization of molecular bands.

20. Ohman, Y. 1941. Stockholm Obs. Ann. 13, No. 11, p.1.

21. Le Borgne, J.F. and Crovisier, J. 1987. Proc. Symp. on Diversity and

Similarity of Comets. Brussels, Belgium.

22. Sen, A.K., Joshi, U.C. and Deshpande, M.R. 1989. Astr. Ap. 217,

307.

The following papers refers to isotopic studies:

23. Altwegg, K. and Bockelee-Morvan, D. 2003. Space Sci. Rev., 106, 139.

24. Bockelee-Morvan, D. 2005. In Comets II, Eds. M.C. Festou, H.U,.

Keller and H.A. Weaver, Univ. Arizona Press, Tucson, P.391.


CHAPTER 6

Gas Production Rates in Coma

The gaseous molecules vaporized from the nucleus are subjected to a va-

riety of physical processes which could dissociate them step by step. Hence,

the species observed in the spectra of comets could comprise contributions

arising out of the dissociated products, as well as those released directly

by the nucleus or even by dust particles. Therefore, the physical quan-

tity of interest is the production rate of the species and its variation with

the heliocentric distance. The production rate of a molecule can be de-

termined either from the observation of the total luminosity or from the

surface brightness of a given spectral line, band or a band-sequence.

If the vapourization of H2O is mainly due to the absorption of the solar

energy, then the production rate of H2O should also have a r−2 depen-

dence with the heliocentric distance. Therefore, if the observations show

a deviation from the r−2 dependence, it is of great significance and has

important information with regard to the mechanism of evaporation of the

material from the nucleus. It is also of interest to see whether the produc-

tion rate of the molecules differs from comet to comet and whether it has

any correlation with the morphology, dust content, age, etc. of the comet.

From these results, one may be able to understand better the evaporation

of material from the nucleus as well as the source of the observed radicals

and molecules in comets and the structure of the nucleus. Some of these

aspects will be the subject of discussion in this chapter.

141



6.1. Theoretical Models

6.1.1. From the total luminosity

The total production rate of a particular species can be determined from

the measurement of the total flux of the radiation from the entire coma of

a given line or band. If the mean lifetime of the molecule is τi then the

coma will contain Ni molecules of this type at any given time so that

Ni = Qiτi, (6.1)

where Qi is the production rate of the species. Since the excitation of the

lines is due to the resonance fluorescence process, for an optically thin case,

the luminosity of any line or a band is proportional to the total number of

the species in the coma and to the fluorescence efficiency factor called the

‘g-factor’. The g-factor actually represents the probability of scattering of

a solar photon per unit time per molecule. The luminosity at wavelength

λi is therefore given by

Li = giNi = giQiτi. (6.2)

Since the observations include the whole extent of the coma, the flux re-

ceived on the Earth Fi is given by

Fi =Li

4π∆2. (6.3)

where ∆ is the geocentric distance. Therefore, the production rate is given

by

Qi =4π∆2Figiτi

(6.4)

The production rate can therefore be calculated provided gi, τi and Fi are

known. In general, the measurements covering only a given instrumental

field of view is available and not over the whole coma. For such cases the

total number of species in the field of view, Ni is given by the relation

Ni =4π∆2Figi

and hence the column density can be calculated.

The g-factor represents the number of photons per second scattered by

a single atom or molecule exposed to the unattenuated sunlight. If the solar

photon flux is πFν per unit frequency in the neighbourhood of the line, the

g-factor for a strict resonance scattering is given by

gν = Fν

∫αν dν (6.5)


Gas Production Rates in Coma 143

where αν is the absorption cross-section. Therefore,

gν = πFν

(πe2

mc

)f (6.6)

or

gλ =

(πe2

mc2

)λ2fλ(πFλ) photons/sec/molecule (6.7)

where fλ is the absorption oscillator strength and πFλ the solar flux per

unit wavelength. For molecules, the relative transition probabilities for

downward transitions must be taken into account. Therefore, Eq. (6.7)

can be written as

gλ =

(πe2

mc2

)λ2(πFλ)fλω (6.8)

where

ω =Av′v′′∑v′′ Av′v′′

.

Here the A’s are the Einstein coefficients. The g-factor depends upon the

oscillator strength and the solar flux. As pointed out in Chap. 3, the solar

radiation in the ultraviolet region is dominated by line emissions. Therefore,

the Doppler shift due to the comet’s heliocentric radial motion can produce

large changes in the g-factor as the comet moves in its orbit (see Fig. 5.14).

This has to be taken into account in the analysis of the observations. But

the average g-factor has been calculated for many molecules and is available

in the literature.

The calculation of the lifetime of the species involves a knowledge of the

photodissociation and photoionization rates, Jd and Ji respectively. These

can be calculated from the relation

Ji,d =

∫ 0

λc

σ(λ)πF(λ)dλ (sec/molecule) (6.9)

where σ(λ) is the photodestruction cross-section and F(λ) is the incident

solar flux. λc represents the threshold for the dissociation or ionization

process. For most of the species, the photodissociation lifetime is relevant

since the dissociation energies are smaller than those of ionization poten-

tials. However, for the case of the CO molecule, the two are comparable

as the dissociation potential ∼ 11 eV and the ionization potential ∼ 14 eV.

The lifetime of the molecules is just the inverse of Eq. (6.9), i.e.,

τi = J−1 sec . (6.10)



It should be noted that the dependence of the solar flux with heliocentric

distance (Eq. 6.9) also enters in the lifetime of the molecules. Usually,

the lifetime is referred to 1 AU. Since both gi and τi depend directly on

the solar flux which varies as r−2, the product giτi [Eq. (6.4)] becomes

independent of r. Because of the presence of large uncertainties in the

value of cross-sections as well as the solar fluxes in the extreme ultraviolet

region, the calculated J values and in turn the lifetime of the species are

also uncertain. For example, the solar flux itself could vary by factors of 2 to

4 during the solar cycle. Therefore, the accurate calculation of the lifetime

of a species is one of the main problems at present. The uncertainty in the

lifetime of the molecules directly affects the calculated production rates.

6.1.2. From surface brightness distribution

It is possible to get much more information than just the production

rates, if the variation of brightness of various emissions with radial distance

from the centre of the comet could be observed. It is a common practice

nowadays to make such observations using a diaphragm whose field of view

is much smaller than the projected size of the coma. A typical observed

brightness profile has a flat region near the centre and falls rather steeply

in the outer region (Fig. 6.1). If the average brightness of a molecule is

denoted as Bi at the projected distance ρ as seen in the sky then

Bi(ρ) = N i(ρ)gi, (6.11)

where N i(ρ) is the average column density of the molecule along the line

of sight. To get the total number of a species one has to integrate over the

model dependent variation of density as a function of the projected distance

ρ from the nucleus. This is then to be convolved with the instrumental field

of view.

It is generally believed that the observed species are the dissociated or

the decayed products of the original complex molecules which were vapour-

ized from the nucleus. Because of the finite lifetimes of these dissociated

products, they in turn break up further as they move out in the coma.

Therefore, the coma can be divided roughly into three zones: a productive

zone, an expansion zone and a destructive zone.

Let us start with the simplest model in which molecules are released by

the nucleus with a production rate

Q = 4πR20E (6.12)



Fig. 6.1 Theoretical surface brightness distributions based on Haser’s model for differentratios of the scale lengths of daughter and the parent molecule. (Adapted from D’Dell,

C.R. and Osterbrock, D.E. 1962. Ap. J., 136, 559).

where R0 is the radius of the nucleus and E is the evaporation rate

(/cm2/sec). It then expands isotropically with a velocity v. For such a

case, the density falls off as R−2 where R is the distance from the centre of

the coma and it is given by

n(R) =Q

4πv

1

R2. (6.13)

At a projected distance ρ as seen in the sky, the column density is obtained



by integrating along the line of sight and is given by

N(ρ) =Q

4v

1

ρ. (6.14)

The column density, therefore, varies as (1/ρ). The above formalism is very

simple and gives only rough estimates.

As the molecules move outwards, they are generally dissociated by the

solar radiation. They have a mean lifetime τ . This corresponds to a distance

Rd = vτ , where v is the average velocity of the molecule. The density

distribution is now given by

n(R) =Q

4πvR2e−(R/Rd),

=C

R2e(−R/Rd) (6.15)

where

C =Q

4πv.

A slightly more complicated model is based on the parent-daughter hypoth-

esis in which the parent molecules released by the nucleus decay with a mean

lifetime τ1 corresponding to a distance (called scale length) of R1 = v1τ1.

The daughter products then in turn decay with a lifetime τ0 corresponding

to R0 = v0τ0. For such a two-component model, the number density takes

the form

n(R) =C

R2

(e−β0R − e−β1R

). (6.16)

Here β0(≡ (τ0v0)−1) denotes the reciprocal of the mean distances travelled

by or scale lengths of the observed molecules, β1 denotes the same by the

parent molecule before they are dissociated. The Eq. (6.16) can be written

as

n(R) =C

R2f(R) (6.17)

where f(R) is the function representing the quantity in brackets of

Eq. (6.16). If n(R1) denotes the density at the distance R at which f(R)

has its maximum value, then

n(R) = n(R1)R2

1

R2

β1

β1 − β0

[e−β0(R−R1) − β0

β1e−β1(R−R1)

]. (6.18)

The integration of the above equation along the line of sight at the projected

distance ρ from the nucleus gives the column density and hence the surface



brightness at that point for an optically thin case. The resulting expression

is of the form

N(ρ) = 2n(R1)R21

β0β1

β1 − β0e+β0R1

1

β0ρ[B(β0ρ)−B(β1ρ)] (6.19)

where

B(z) =π

2−∫ z

0

K0(y)dy. (6.19a)

Here K0(y) is the modified Bessel function of the second kind of order zero.

The Eq. (6.19) can be written as

N(ρ) = (constant)s(x) =(constant)

x

[B(x)−B

(β1x

β0

)](6.20)

where x = β0ρ. The equation shows that the shape of the surface brightness

distribution s(x), depends essentially upon the relative values β1 and β0.

The Fig. 6.1 shows a plot of brightness profiles predicted for various values

of the ratio β1/β0. The curves become rather flat in the central region for

small values of β1/β0, and this means that the two scale lengths do not

differ from each other very much. One can also see the deviation from the

simple model (constant outflow velocity with no creation or destruction) in

which the dependence is given by 1/ρ.

The total number of molecules M(ρ) within a cylinder of radius ρ cen-

tred on the nucleus can be obtained by the integration of Eq. (6.19) over

ρ, i.e.,

M(ρ) =

∫ ρ

0

2πσN(σ)dσ

= 4π n(R1)R21

β1

β0(β1 − β0)eβ0R1β0ρ[G(β0ρ)−G(β1ρ)] (6.21)

where

G(z) =π

2−∫ z

0

K0(y)dy +1

z−K1(z).

Here K0 and K1 are the modified Bessel functions of the second kind

of order 0 and 1 respectively. The above expression can also be expressed

more conveniently in terms of the production rate as

M(ρ) =Qρ

v

[∫ µx

x

K0(y)dy +1

x

(1− 1

µ

)+K1(µx)−K1(x)

](6.21a)

where x = β0ρ and µ = β1/β0. The quantity occurring in bracket depends

only on the parameters µ and x which could be tabulated for various values

of x and µ.



The above two-stage model usually known as Haser’s model, can be

further extended to three stages in which the daughter products themselves

dissociate. Let τ1, τ2 and τ3 denote the lifetime of the parent, daughter and

the granddaughter species respectively. If the corresponding velocities are

v1, v2 and v3 then βi = (τivi) with i = 1, 2 and 3. The densities of the

three corresponding species can be expressed conveniently in terms of the

production rate Q1 of the parent molecule as follows.

n1(R) =Q1

4πv1R2exp(−β1R) (6.22)

n2(R) =Q1

4πv2R2

β1

(β2 − β1)[exp(−β1R)− exp(−β2R)] (6.23)

and

n3(R) =Q1

4πv3R2[A exp(−β1R) +B exp(−β3R) + C exp(−β2R)] (6.24)

where

A =β1β2

(β1 − β2)(β1 − β3),

B =−A(β1 − β3)

(β2 − β3)

and

C =−B(β1 − β2)

(β1 − β3).

The corresponding column density of the species over the field of view of

linear radius ρ is given by

N1(ρ) =Q1

4πv1

2

ρB(β1ρ), (6.25)

N2(ρ) =Q1

4πv2

2

ρ

(β1

β2 − β1

)[B(β1ρ)−B(β2ρ)] (6.26)

and

N3(ρ) =Q1

4πv3

2

ρ[AB(β1ρ) +BB(β2ρ) + CB(β3ρ)] (6.27)

where B(Z) is given by Eq. (6.19a). The above procedure can easily be

extended to cascade processes involving more than three species.



6.1.3. From number densities

The relation between the emission coefficient j and the number density

n of the species for the resonance fluorescence excitation process is given

by

j

hνem=

π e2f

hνabsmρν(r)nP. (6.28)

Here νem and νabs are the emission and absorption frequencies, ρν(r) the

solar radiation density at the heliocentric distance r and P is the probability

that a particular transition takes place in comparison with other decays. All

other constants have their usual meanings. The integration of Eq. (6.28)

along the line of sight through the comet and up to a projected radius ρ

gives the relation

L(ρ)

hνem=

πe2f

hνabsmρν(r)PM(ρ) (6.29)

where L(ρ) is the observed luminosity in the emission line and M(ρ) is the

total number of the species in the field of view. The above equation can be

written as

M(ρ) =

(νabs

νem

)mL(ρ)

πe2fρν(r)P(6.30)

or

log M(ρ) = logC + log L(ρ) + 2 log r (6.31)

where

C =

(νabs

νem

)m

πe2fρν(r)P.

The last term in equation (6.31) arises as solar radiation varies as r−2

where r is the heliocentric distance of the comet. The constant C depends

upon the transition of interest.

For C2 molecule, as one generally observes band sequence fluxes with

∆V = +1 or 0, it is necessary to include in the calculation of constant

C, the absorptions arising from various vibrational levels from the lower

electronic state through the term (fρν(r)/ν). In a similar manner the

population of vibrational levels in the upper electronic state enters through

the probability P . Therefore, a mean value for the above ratio has to be

used, i.e., (fρ

ν

)=∑v′′

xv′′∑

fv′′,∆v ρ(νv′′,∆v)/νv′′,∆v. (6.32)



Here v′′ represents the vibrational level in the lower electronic state and

∆v the various band sequences being considered. xv′′ represents the weight

factors for the vibrational levels of the lower electronic state. Similarly the

transition probability P for the downward transition should also take into

account the population distribution in the upper vibrational levels. The

Eq. (6.31) gives the abundance of the species of interest within the field of

view of the comet as a function of r.

The comparison of the observed surface brightness distribution with the

expected distribution of Fig. 6.1, fixes the values of β0 and β1 as well as

R1 from its definition. These can then be used to compute M(ρ) from

Eq. (6.21) for comparing with the observed distribution [Eq. (6.31)]. This

fixes the value of n(R1) and hence the density distribution in the coma

through Eq. (6.18). The observed distribution of M(ρ) can also be used to

derive the production rate Q directly from Eq. (6.21a).

The radial outflow decay model has been used extensively in the litera-

ture with a high degree of success. Basically from a fit of the expected sur-

face brightness distribution with the observed distribution, it is possible to

derive the production rates and the two scale lengths corresponding to the

parent and the dissociated product. From a knowledge of the scale lengths,

the lifetime of the species can be determined provided the velocity is known.

However, it is found that the scale lengths derived from the Haser’s model

are generally smaller than the values calculated using the estimated out-

flow speed of the gas and the photodissociation lifetime computed from the

solar UV flux and the measured photoabsorption cross-sections. There are

some limitations of this model. First of all it is applicable only to pho-

todestruction products. It does not allow for situations where they could

be produced through chemical reactions. Other effects like non-radial flow,

the effect of solar radiation pressure and the velocity distribution of the

species have to be taken into account in a more realistic model. Some at-

tempts have been made in this direction. Mention may be made of the

method based on the vectorial formalism. In this method, the molecular

fragments of a dissociated parent molecule are ejected isotropically in a

reference frame attached to the parent molecule. Also, unlike in the case

of Haser’s model, the effects of velocity and lifetimes are separated which

allow the study of velocity dependent phenomena. These are the two main

improvements over the Haser’s model. Another formalism has been devel-

oped based on an average random walk model which is the Monte Carlo

approach. The calculations are done in 3-dimensions centred on the comet

nucleus. The results are then reduced to 2-dimensional maps for compar-



ing with the observation of molecules. It allows the asymmetric ejection of

the parent molecules from the molecules such as jets etc. It can also take

into account the acceleration due to solar radiation pressure. It includes

several dissociation steps for each specie. More physical effects could be

incorporated into the model. These two approaches are more complicated

than Haser’s model. However, these models yield the surface distributions

which resemble rather closely with those of Haser’s model.

6.1.4. Semi-empirical photometric theory

The gas production rate can, in principle, be determined from the knowl-

edge of the observed light curve. The light curve of a comet basically gives

the variation of apparent brightness as a function of the heliocentric dis-

tance. In general, the observed brightness in the visual region is mainly due

to the continuum and the Swan bands of the C2 molecule. The continuum

is made up of scattering by the dust particles in the coma as well as the

reflection from the nucleus. Therefore, a simple photometric equation can

be written relating the observed light to the light contributed by the dust

grains and the nucleus in the same visual band pass. It can be written as

Aφn(α) +Bf1(r) + Cφd(α)f2(r) = r2100.4[m−m1(r)−5 log ∆]. (6.33)

The three terms on the left hand side of the above equation represent the

contribution from the nucleus, gas and dust respectively. Here m1(r) is

the apparent visual magnitude of the comet, m the apparent magnitude

of the Sun, φn(α) and φd(α) are the phase functions of the nucleus and

the dust, at phase angle α, f1(r) and f2(r) are the functional behavior of

the gas and dust production with r. The equation (6.33) can be simplified

with the assumption that the cometary activity is basically given by the

production rate of hydrogen QH . This is a reasonable assumption as H2O

is the most abundant molecule in a comet and H is the dissociated product

of H2O. If the gas and dust are well mixed and remain almost the same

with several apparitions, then one can write

Cf2(r) ' C(r)QH(r). (6.34)

The observed production rate of C2 of several comets appears to vary

roughly as a quadratic function of QH . Therefore, one can write to a

first approximation,

Bf1(r) ≈ B(r)Q2H(r). (6.35)



Therefore, Eq. (6.33) can be written in the form

PnR2nφn(α) +B(r)Q2

H(r) + C(r)QH(r)φd(α) = r2100.4[m−m1(r)−5 log ∆]

(6.36)

where Rn and Pn represent the radius of the nucleus and geometric albedo

respectively. The Eq. (6.36) can be solved as a quadratic equation inQH(r).

For further simplification one can write the ratio of the dust scatter to that

of the gas in the same band pass as

δ =C(r)QH(r)φd(α)

B(r)Q2H(r)

. (6.37)

The term B(r) should be proportional to the local lifetime of the molecule

and so it can be written as

B(r) = Rτc2(r/r0)2 (6.38)

with the C2 lifetime referring to r = r0 = 1AU. Here R refers to the

resonance fluorescence efficiency. The Eq. (6.36) can be written as

PnR2nφn(α) + (1 + δ)Rτc2 Q

2H(r) = r2100.4[m−m1(r)+5 log ∆]. (6.39)

This gives the solution for the hydrogen production rate as

QH(r) =

[r2100.4(m−m1(r)+5 log ∆) − PnR2

nφn(α)

(1 + δ) Rτc2

]1/2

. (6.40)

If the second term involving PnR2n is smaller than the first term, which is

the case at shorter distances, it is easier to calculate QH from the observed

light curve.

6.2. Results

6.2.1. OH and H

Extensive observations of the Lyman α of hydrogen as well as of OH

emission at 3090 A have been carried out for various comets using rockets

and satellites. These observations have been used to extract the produc-

tion rate of OH and H. If OH and H arise from the dissociation of H2O,

the ratio is Q(H)/Q(OH) ≈ 2. The analysis of the brightness measure-

ments for the Comet Bennett showed that the dependence of the produc-

tion rates of H and OH with heliocentric distance had almost the same

variation as r−2.3 (Fig. 6.2). This implies that both H and OH must have

come out of the same parent molecule, which presumably is water. The



Fig. 6.2 A plot of the production rates of OH and H as a function of heliocentricdistance for Comet Bennett (Keller, H.U. and Lillie, C.F. 1974. Astr. Ap., 34, 187).

derived production rates for OH and H at 1 AU were (2.0 ± 0.8) × 1029

molecules/sec and (5.4 ± 2.7) × 1029 atoms/sec respectively. This gives a

ratio of Q(H)/Q(OH) ≈ 2.7. The International Ultraviolet Explorer Satel-

lite (IUE) was well suited for this study as it covers the wavelength region

from 1150 to 3400 A wherein the emissions from H and OH lie (Fig. 4.4).

Therefore, it was possible to make simultaneous brightness measurements

of these decay products as a function of the radial distance from the nu-

cleus. Such spectra covering a wide range of heliocentric distances give a

homogeneous set of observations. From a fit with the expected brightness

distribution with those of observations, the production rate of H2O can

be determined for various heliocentric distances. The results for Comet

Bradfield indicate the dependence of the production rate with heliocentric

distance as r−3.8 to r−4.4. For Comet Austin (1990 V) the variation is

around r−1.8 to r−2.8. The ranges in the power of r for a given comet come

about due to the fact whether the effect of solar activity is included or not

in the model calculations. The derived production rates of water in Comet

Bradfield can also explain reasonably well the observed brightness distri-



bution of hydrogen and oxygen atoms with two velocity distributions for H

atoms of 8 and 20 km/sec (Sec 6.3.1). Therefore, the observed brightness of

OH, H and O are consistent with a common source of all the three species,

which most probably is H2O. The heliocentric variation of the production

rate of OH for Comet Stephan-Oterma is found to have a dependence, as

r−4.8±0.6 for the pre-perihelion distance. The heliocentric variation of OH

production rate in Comet Hale-Bopp for pre-perihelion observations is ini-

tially steeper (r−3.7), then flatter (r−1.8) and finally much steeper (r−6.8).

The typical number densities of H2O and OH at a distance of 104 km from

the nucleus are of the order of 2× 105/cm3 and 4× 104/cm3 respectively.

Comet Halley was observed extensively from IUE satellite for Ly α of

H emission (1216 A) and OH emission (3085 A). The observations of Ly α

were analysed based on Monte Carlo models in combination with radiative

transfer models. The derived production rate of water from the nucleus

centred observations is compared with that derived from OH observations

in Table 6.1. The agreement in the production rate of H2O derived from H

and OH observations is good.

Table 6.1 Comparison of H2O production rates

for Comet Halley∗.

Log Q(H2O) Log Q(H2O)

(r AU) (from OH, 3085 A) (from Ly α, H)

1.45 28.987 28.978

1.13 29.582 29.5091.02 29.555 29.452

0.84 29.818 29.734

0.98 29.642 29.6911.32 29.367 29.343

1.85 29.340 29.406

0.84a 29.36 29.45a

*IUE observations.Adapted from Combi, M.R. and Feldman, P.D.1993. Icarus, 105, 557.a. For Comet Hyakutake: Combi, M.R. et al.1998. AP.J., 494, 816.

The study of the 18 cm line of OH in the radio region should also

give information about the production rates and the velocity fields in the

coma. This should supplement the information obtained from the study



of the lines in the ultraviolet region. A large amount of work has been

carried out on various comets based on the radio frequency lines of the OH

radical. Since the radio frequency lines are very narrow and since it is also

possible to make high resolution radio observations, it is possible to study

the velocity distribution of OH molecule in the coma. In fact, the precision

of the velocity field is limited only by the quality of the data. The observed

profile is generally asymmetric about the centre. In addition, there is an

asymmetry in the spatial brightness distribution in the East-West of the

comet’s centre. These asymmetries arise due to the fact that the velocities

of the OH molecules in the coma are not uniform throughout, i.e., the

differential velocity effect is present. The observed asymmetry implies that

the OH radicals are not isotropically distributed around the nucleus.

The calculation of the total number of OH molecules from the observed

emission is not so simple as the lines arise due to the excitation by ultravi-

olet radiation where Swings effect is very effective. The expression relating

the total flux density emitted by the entire comet to the total number of

OH molecules in the ground state is given by

F =Aul ikTBG

4π∆2

2Fu + 1

8NOH (6.41)

where TBG is the background emission, Fu is the total angular momentum

quantum number of the upper state of the transition, i is the ‘inversion’

of the lambda doublet (Fig. 5.14) and ∆ is the geocentric distance of the

comet.

The OH production rate can be calculated from the relation QP =

NOH/τOH where τOH is the lifetime of the OH molecules. In general, for a

better estimate it is necessary to take into account the collisional quenching

as well as the distribution of molecules and the molecular outflow veloci-

ties in the coma in the interpretation of the observed flux in terms of the

total number of molecules and hence the production rate. To specify the

distribution of OH in the coma requires a model for the production of OH

from H2O and the destruction of OH due to photodissociation and pho-

toionization. The Haser’s model is most commonly used for the analysis of

OH observations, i.e.,

n(OH) =QP

4πr2vOH

lOH

lOH − lP

(e−r/lOH − e−r/lP

)(6.42)

where n(OH) is the density of the OH molecule. lOH and lP are the scale

lengths (same as β−1 of Sec. 6.1.2) for OH and the parent molecule re-

spectively. vOH is the radial velocity of the OH molecule. Figure 6.3 shows



Fig. 6.3 Shows the production of OH as a function of the total visual brightness reducedto r = ∆ = 1 AU for various comets. The dashed line drawn is for logQp=30.33 - 0.28

m1 (Despois, D., Gerard, E., Crovisier, J. and I. Kazes, I. 1981. Astr. Ap., 99, 320).

a plot of the OH-parent production rate as a function of the total visual

brightness for a number of comets, both referred to r = ∆ = 1 AU. The

dashed line represents the relation

log QP = 30.33− 0.28m1. (6.43)

It is of interest to compare the production rates of OH obtained from both

the radio and the ultraviolet observations for the same comet, to see whether

there is any discrepancy between the two determinations. It is found that

the ultraviolet production rates are systematically higher compared to those

of radio values.

There are several causes for this observed discrepancy. First of all the

two measurements are fundamentally different and hence comparison relies

on extensive modelling. It requires the knowledge of lifetime of OH and the

velocity distribution. The model also depends upon the heliocentric dis-

tance, radial velocity, anisotropic out-gassing and the solar cycle. Owing

to all these factors, the problem is a complex one. Therefore, the discrep-

ancy appears to be associated with the use of different model parameters

which enter in the interpretation of the UV and radio observations. For



example, the lifetime of OH used in the two cases are different. In addi-

tion, Λ-doublet OH population is sensitive to collisions in the inner coma

and so it has to be taken into account. Therefore, the derived production

rate of OH from radio observations depends upon the beam width of the

radio telescope and so on. Basically, the physics of the problem is com-

plicated and it involves the specification of too many parameters. So far

a comparison of the production rate of OH from UV and radio methods

were carried out for different dates of observations, radio telescopes and

so on. Therefore, in order to understand better this discrepancy, if it is

real, is to use simultaneous UV and radio observations so that the physical

characteristics of the coma are roughly the same. Through this procedure

it may be possible to put tight constraints on the model parameters. Such

an exercise was carried out for Comet Halley observations of 1985-86. The

OH observations from IUE satellite and 18 cm radio observations of Comet

Halley taken around the same time have been used for the study. The 18 cm

data has been analysed with sophisticated models. The derived production

rate of OH from OH observations and 18 cm radio observations are given in

Table 6.2. The agreement between the radio and UV OH production rates

is good.

Table 6.2 OH production rates

from UV and radio observations of

Comet Halley.

Date Method Q(OH)

(1985) (1029/cm2)

2.5 Dec UV 1.4

3.2 Dec UV 1.015.5 Dec UV 1.7

5.12 Dec Radio 1.4

10-12 Dec Radio 1.2

OH UV (0, 0 band of (A−X)3085 A), Radio (18 cm)Adapted from Gerard, E. 1990. As-tron. Astrophys., 230, 489.

The study of 18 cm line of OH has been carried out routinely on comets.

The results for Comet Halley showed a variation of the production rate with

the heliocentric distance as r−2 for r < 2 AU.



The production rate of H2O can also be determined independently of

the study of [OI] 6300 A and H α (6565 A) observations of comets. This will

provide an independent check on the consistency of O and H arising out of

H2O. Table 6.6 shows the derived values for the production rate of H2O for

Comet Halley, by these two methods based on the observations carried out

on the same night and with the same instrument.The agreement between

the two is good. Therefore, the production rates of H2O derived from the

Hα brightness measurements are consistent with the H2O production rates

derived from the O[1D] measurements, both of which are based on the coma

containing OH, H and O arising out of the dissociation of H2O. Therefore,

at the present time, it is an excellent approximation that H2O is the source

of observed species OH, H and O in comets.

The heliocentric variation of the production rate of H2O is of particular

interest. As already mentioned, if the evaporation of the gases from the

nuclei is due to the absorption of the total solar energy, the variation of

the production rate of H2O should also behave as r−2. The results pre-

sented earlier showed that for some comets it is in reasonable accord with

this dependence. However, for some other comets the variation is vastly

different. This is related to the evaporation of the gaseous material from

the nucleus which is dependent upon the nature and the composition of the

nucleus. Therefore, more complicated models might have to be considered

for explaining the observations.

6.2.2. H2O, H2

H2O

2.7 µm band

Before the direct detection of H2O in Comet Halley, the presence of H2O

in comets was inferred indirectly from observations such as the presence of

OH, H, O and H2O+ in comets. The fundamental bands of vibration, es-

pecially the ν3 near 2.7 µm cannot be observed from ground due to strong

absorption by the Earth’s atmosphere. The first direct detection of ν3 band

at 2.7 µm of H2O came from the observations carried out with KAO on

Comet Halley at r = 1.13 AU (Fig. 4.7). It was also detected by VEGA

spacecraft. This detection provided direct evidence that H2O is the domi-

nant volatile component of comets. The total column density of H2O can be

estimated from the measured band brightness and with the use of Haser’s



model. It is then possible to calculate the production rate of H2O. The

calculated production rate of H2O ∼ 4 × 1028/sec. The 2.7 µm band of

H2O has been observed in other comets like Wilson, Hale-Bopp, Hartley

etc.

Radio lines

The pure rotational transitions between low rotational states of wa-

ter are predicted to be quite strong. These lines lie in the submillimetre

wavelength region and can only be observed from space. The line at 557

GHz arising out of 110 − 101 transition in H2O was detected in Comet Lee

(r ∼ 1.2 AU) using Submillimetre Wave Astronomy Satellite (SWAS). The

estimated production rate of H2O from the observed intensity and use of

Haser’s model ∼ 8× 1028/sec.

Prompt emission lines

The photolysis of water produces OH mostly in the excited states of the

ground electronic state X2Π as shown by laboratory studies. The wave-

length of vibrational rotational transitions of (1, 0) band arising out of the

ground electronic state X2Π lie in the infrared region (∼ 3 µm). The tran-

sitions from v′′ = 1 to v′′ = 0 occurs within ∼ 10ms. This type of emission

is called prompt emission lines which is similar to the ones discussed earlier

which lie in the UV region (Sec. 5.1.5).

The prompt emission lines arising from the vibrational rotational transi-

tions of (1, 0) band in the ground electronic state X2Π is shown in Fig. 6.4.

The four lines in the region 3042 cm−1 to 3048 cm−1 are conspicuous.

Many of the nonresonant fluorescent bands of H2O called hot bands

which lie in the wavelength region of around 2.9 µm can be observed from

the ground. Therefore with the same instrument it is possible to make

simultaneous measurements of the parent (H2O, hot bands) and its disso-

ciation product (OH, prompt emission lines) in the infrared region.

If H2O is the parent molecule of OH prompt emission lines, the spatial

distribution of their intensities should be similar, peaking at the centre in

contrast to quiescent OH which should exhibit a much flatter distribution.

Figure 6.5 shows the observed intensities of H2O and OH as a function of

distance from the nucleus in Comet Lee. They are in good agreement with

each other. This type of agreement has been seen from several comets.

The production rate of H2O can be calculated from the observed inten-



Fig. 6.4 High dispersion spectra of Comets Lee and LINEAR showing quodruplet lines

of OH near 3046 cm−1 (Bonev, B.P., et al. 2004. Ap.J., 615, 1048, Reproduced bypermission of the AAS).

sity of its nonresonant fluorescence lines (hot bands) from the relation

Q(H2O) = P

⟨F(H2O)

g(hcν)

⟩(GF) (6.43a)

Where F(H2O) is the line flux and g is the fluorescene emission factor. P is

a parameter which depends upon geocentric distance, the photodissociation

lifetime of H2O and the fraction of H2O molecules expected in the sample

region of the coma. The quantity (GF) represent the growth factor that

accounts for the loss of flux near the nucleus.

The equivalent g-factor geq for OH prompt emission lines which equal

the observed production rate of H2O represented as Q∗ can be calculated

from an expression similar to equation (6.43a) as

Q∗ =F(OH∗)

geq(hcν)∗(GF)∗

Here all the parameters are similar to those of equation (6.43a), but refer

to OH prompt emission lines. These geq factors can be used to derive the

production rate of H2O in any other comet from the measured fluxes of



Fig. 6.5 Observed Spatial distribution of H2O and OH in Comet Lee (Bonev, B.P. et

al. 2004. Ap. J., 615, 1048, Reproduced by permission of the AAS).

OH prompt emission lines. This is a simpler way of getting the production

rate of H2O in comets as it is easier to observe OH prompt emission lines

from the ground. Therefore OH prompt emission lines provide a proxy for

water.

The distribution of OH prompt emission lines seen in X2Π, v′ = 1 state

in comets also refer to the conditions which is vastly different from those

carried out under laboratory conditions. Therefore cometary observations

of OH prompt emission lines could provide deeper insight into the dissoci-

ation dynamics of H2O.

Total water production

The production rate of H2O for Comet Halley has been measured by

various methods over a wide range of heliocentric distances covering before

and after perihelion passage. Integration of the observed production rate

of water with time gives an estimate of the total amount of water subli-

mated from the nucleus during the entire apparition, which comes out to be



1.3×1011 kg. For Comet Borrelly, the estimated total amount of water sub-

limated from the nucleus obtained from integrating the water production

rate throughout an apparition is around 1.1× 1010 kg.

H2

The strong lines of the hydrogen molecule should appear in the spectral

region around 1600 A. But its identification is not definite. Shorter than

around 1170 A, three H2 Lyman P1 lines of the (6, v′′) bands have been

detected in Comet LINEAR with FUSE (Fig. 4.6). These lines are normally

not excited but arises due to accidental coincidence of the wavelength of

solar Lyman β line with the wavelength P1 line of (B 1Σ+u− X 1Σ+

g ) (6,

0) band. The derived column density of H2 of (3.0 ± 0.6) × 1013/cm2 is

found to be consistent with the H2O dissociation models. i.e. H2O → H2

+ O(1 D). It is possible that some of it could be produced directly from

the nucleus or by solar wind sputtering of dirty ice grains.

6.2.3. CN, C2, C3, NH

It is easier to study the spectral lines of the molecules like C2, CN, etc in

comets using the standard techniques as their lines lie in the visual region.

Generally filters are used which can isolate their emission bands. It is also

possible to get a homogeneous set of data on many comets covering a large

range of heliocentric distances which are of prime importance for investi-

gating the possible physical correlations like the composition variations or

the similarities among the comets of various types.

The method based on Eq. (6.31) is particularly suitable for the study

of molecules like C2, CN, etc., as the total flux of a band or a band se-

quence in a certain field of view can easily be measured using the standard

methods. The total number of the molecules can then be calculated from

Eq. (6.31) as the constant can be evaluated using the spectroscopic data

for the molecules of interest. The production rate can be calculated from

Eq. (6.21a) from molecular column densities knowing the scale length of

the two species and for an assumed outflow velocity, ∼ 1 km/sec. The

production rates can easily be scaled for any other value of the outflow ve-

locity. The generally used scale lengths for some of the molecules are given

in Table 6.3. The results for the heliocentric variation of the production

rates of C2, CN and C3 for Comet Bradfield show a dependence as r−3.2

compared to r−1.7±0.3 for CN observed for Comet West. For the dusty



Comet Stephan-Oterma, the production rate of CN and C2 varied approx-

imately as r−4.2±0.3 and r−5.6±0.4 respectively. Therefore the heliocentric

variation of the production rate of molecules vary among comets.

Table 6.3 Scale length of molecules at 1 AU.

Molecule Parent(km) Daughter(km) Daughter

life time (sec)

OH(0,0) 2.4 × 104 1.6 × 105 1.6 × 105

NH(0,0) 5.0 × 104 1.5 ×105 1.5 ×105

CN(∆v = 0) 1.3 × 104 2.1 ×105 2.1 × 105

C3(λ4050A) 2.8 × 103 2.7 ×105 2.7 ×104

C2(∆v = 0) 2.2 ×104 6.6 ×104 6.6 ×104

Schleicher, D.G. and Farnham, T.L. 2005, In Comets II, eds.M.C. Festou, H.U. Keller and H.A. Weaver, Univ. Arizona

Press, Tucson, p. 449.

Several studies have been carried out in a systematic manner on a large

number of comets by various groups. One such study contains 85 comets. A

statistical study of the variation of the production rates of several molecules

has been studied.

Comet Halley was observed extensively during its apparition in 1985-

86. Therefore extensive good quality data exists on this comet covering a

wide range of heliocentric distances, before and after perihelion passage.

The results for the heliocentric variation of the production rates of several

molecules is shown in Fig. 6.6. The dependence of production rate with

heliocentric distance for the molecules C2,C3 and CN are quite similar. The

slope of NH is also quite similar to those of carbon bearing species. Another

feature of Fig. 6.6 is the presence of asymmetry about perihelion in the

production rates of these species.

The curve of OH varies gradually with heliocentric distance and also has

slight asymmetry about perihelion. The possible difference in the variation

in the production rates with heliocentric distance between the minor species

and water, the major constituent, has also been seen in several other comets.

This appears to indicate decoupling of carbon-bearing species from that of

water.

It is of interest to look at the relative abundances of the species among

comets. The average relative molecular production rates of several species

for Comet Halley is given in Table 6.4. The mean values and the range



Fig. 6.6 Observed production rates of molecules OH, NH, CN, C2 and C3 and A(θ)fρ

plotted as a function of the heliocentric distance for Comet Halley. Filled symbols referto preperihelion observations while open symbols refer to postperihelion observations.

Solid lines and dotted curves are for linear and quadratic fit to the data. (Schleicher,

D.G, Millis, R.L. and Birch, P.V. 1998. Icarus, 132, 397).

among all the comets studied in a homogeneous group of 85 comets is also

given in Table 6.4. The observed ratios for Comet Halley fall within the

typical range of values.

Observations for Comet Borrelly are available over several apparitions.

The derived gas production rates of carbon-bearing molecules are larger

before perihelion compared to that after perihelion for the same distance.

The dependence of production rate with heliocentric distance is similar for

CN, C2 and C3 (∼ −8.1) while OH has a dependence ∼ −8.9. The values

are quite similar over different apparitions. The relative abundances of the

species for some comets is given in Table 6.4. There are several other comets

which have either strong asymmetric or steep heliocentric production rate



Table 6.4 Abundance ratios in comets.

Species Wild2e Halleyb Borrellyd Wirtenc Meana Rangea

NH/OH -2.35 ±.34 -1.86 -2.28±.34 -2.36±.03 -2.37±.23 -2.77 to -1.80CN/OH -2.60±.14 -2.32 -2.62±.09 -2.50±.12 -2.50±.18 -2.83 to -2.17C2/OH -2.97±.23 -2.21 -2.91±.12 -2.49±.12 -2.44±.20 -2.90 to -2.10

C3/OH -3.40±.20 -3.20 -3.55±.19 -3.49±.12 -3.59±.27 -4.26 to -3.09

UV Cont. -25.53±.32 -25.53±.31Blue Cont. -25.40±.13 -25.53±.36

Green Cont. -25.28±.24 -25.0 -25.40±.31

C2/CN +0.11 +0.06±.0 -0.09 to +.29

Continuum is in A(θ)fρ/Q(OH).

(a) A’ Hearn, M.F. et al. 1995. Icarus, 118, 223.(b) Schleicher, D.G., Millis, R.L. and Birch, P.V. 1998. Icarus, 132, 397.

(c) Farnam, T.L. and Schleicher, D.G. 1998. Astron. Astrophys., 335, L50.

(d) Schleicher, D.G., Woodney, L.M. and Mills, R.L. 2003. Icarus, 162, 415.(e) Farnam, J.L. and Schleicher, D.G. 2005. Icarus, 173, 533.

dependence such as, Arend Rigaux, Charyumov - Gerasimenko, Giocobini

- Zinner, Kopff and Tempel 2. There are also some comets in which the

behaviour of production rates of species remain roughly the same with

different apparitions and in other comets it changes from one apparition to

another. So there is a wide variation among comets.

On the other hand, Comet Wild 2 shows no asymmetry in their pro-

duction rates between pre - and post perihelion passages and heliocentric

distance. This indicates that C2/CN production rate remained the same

with apparition. So Comet Wild’s composition remained constant over time

and heliocentric distance.

The different properties seen from comets such as asymmetry in the gas

production around perihelion and its variation with heliocentric distance

etc. could arise due to seasonal effect of sources from a complex rotating

comet nuclei which depends upon pole orientation, its stability, direction

of rotation etc.

CN from dust

The detection of CN jets for the first time in Comet Halley indicated

the presence of an extended source of CN arising from the grains. This

conclusion is based on the fact that the radical CN profiles should be peaked

at some distance from the nucleus in contrast to the parent molecule which

should be peaked at or near the nucleus. This is similar to the extended

source of CO and H2CO observed from Comet Halley. The statistical study



of 85 comets showed a good correlation between dust to gas ratio with CN

to gas ratio in several comets. It also indicated that comets with high dust

showed a better correlation supporting the suggestion that dust could be

the dominant source of CN in dusty comets. The observed production rate

also indicated that this effect is quite weak for C2 and C3 and hence their

contribution from grains must be small. The variation of dust production

rate with CN production rate shows a good correlation between the two

indicating that CN originates largely from volatile grains in the coma.

The typical production rates at 1AU for CN, C2, H, OH and H2O ex-

pressed in the unit of Q(H2O)=1 are roughly the following:

Q(H2O) ≈ Q(OH) ≈ 1;

Q(H) ≈ 2;

Q(C2), Q(CN) ≈ 0.01

These values indicate that the production rates of C2 and CN and other

species seen in the visual spectral region are less by a factor of about 100 or

so compared to that of H2O or H. Thus, even though the lines of C2, CN,

etc., dominate the visual spectral region, they are only minor constituents of

the cometary material. Therefore, the total amount of gas ejected from the

nucleus can be represented to a very good approximation by the production

rate of H2O.

6.2.4. CH, NH2

The column density of CH for the Comet Kohoutek at r = 0.5 AU is

about 4×1010/cm2 averaged over 5×104 km. Similarly the column density

of NH2 at r=1 AU is about 6 × 1010/cm2 which is the average value over

3×104 km. These two molecules have also been observed in the Comet West

for the heliocentric distances from 0.60 to 1.58 AU and only the relative

values with respect to C2 are known.The mean relative value of CH and

NH2 with respect to C2 integrated over the whole coma is determined to

be 0.013 ± 0.003 and 0.030 ± 0.013 respectively for distances r <∼ 1 AU.

These values show that the abundance of CH is approximately 1 to 2% of

C2 and that of NH2 about 3% of C2. The derived production rate of CH

and NH2 for Comet Halley at r = 0.89 AU is (4.1 ± 1.2) × 1027/sec and

(1.6±0.3)×1027/sec respectively. This gives a ratio for the production rate

of CH and NH2 with respect to H2O of about 0.007 and 0.003 respectively.



6.2.5. CO, CO2

CO

The ultraviolet observations made on the Comet West at the heliocentric

distance of 0.385 AU showed the strong bands of (A-X) transitions (near

1500 A) of the CO molecule. The deduced production rate of CO from this

observation based on Eq. (6.4) with τ(at 1 AU) = 1.4 × 106 sec comes out

to be

Q(CO) = 2.6× 1029/ sec .

Since then the above bands of CO has been seen from all bright comets

observed with IUE, HST etc.

The derived production rate of CO relative to H2O in Comet West was

around 0.3 and in Comet Bradfield the ratio was about 0.02. Comet Levy

gave a ratio of 0.11±0.02. Comet Halley gave a ratio of ≈ 0.17. For several

comets belonging to Oort cloud comets, the total abundance of CO varied

from 1% to 24% relative to water.

Extended source of CO

The presence of an extended source of CO in Comet Halley came from

the in situ observations carried out with Neutral Mass Spectrometer (NMS)

on board the Giotto spacecraft. This result came from a comparison of the

observed distribution of molecules of m/q = 28(CO) with m/q = 18(H2O).

The observed radial density distribution of H2O and CO derived from NMS

data is shown in Fig. 6.7. The two molecules show entirely different varia-

tions. The linear decrease of H2O is due to the photo-destruction of H2O.

For CO, the initial linear increase is due to the extended source of CO

which is likely to arise from CHON dust particles containing H2CO which

photodissociates. The nearly constant variation at large distances is due to

the long life of CO against photodestruction.

CO2

The presence of CO+2 ion in comets implied the presence of CO2. But

CO2 was observed for the first time in Comet Halley from the infrared spec-

trometer on board the Vega spacecraft which detected ν3 fundamental band

at 4.26 µm. Although the band is strong, it cannot be observed from ground

due to the absorption of CO2 in the earth’s atmosphere. Subsequently it

has been observed in several other comets such as Hale-Bopp, Hartley etc.



Fig. 6.7 Radical density profiles for H2O and CO derived from gas mass spectra with

NMS on Giotto spacecraft for Comet Halley. The solid line corresponds to a least squarefit to the data and corresponds to a photodestruction rate of kH2O ≈ 1.7 × 10−5s−1.

The dashed lines refer to photodestruction rate for ±10%. (Eberhardt, P. 1999. Space

Sci. Rev., 90, 45).

by ISO. The presence of mass peak at 44 in the Giotto NMS mass spectra

also showed the presence of CO2 in Comet Halley. The derived abundance

ratio of CO2/H2O is around 3–4% for Comet Halley and 8–10% for Comet

Hartley 2. The derived production ratio of CO2 to water in comets varies

around 3–10%. However for Comet Hale-Bopp the estimated production

ratio of CO2 to water at r > 2.9 AU was > 20%. Therefore CO2 is also a

trace constituent of the nucleus.



Prompt emission

The Cameron band system of CO seen in comets arises between triplet

and singlet electronic states (a3Π−X1Σ). These transitions are forbidden

dipole ones. Therefore they cannot be excited by resonance fluorescence

process. They are most likely to arise from the formation of CO molecules

in the a 3Π state during the photodissociation of CO2. The molecules in the

a3Π state then decay to the ground state in a time scale of ∼ 7× 10−3 sec

producing prompt emission lines. Therefore the Cameron band emission

is directly proportional to the production rate of CO2 and can be used to

trace the abundance of CO2.

Two dominant mechanisms for the observed Cameron emission are the

photodissociation of CO2 by solar radiation and the electron impact ex-

citation of CO. The two processes can be distinguished observationally as

the rotational temperature of CO molecules produced in the photodissoci-

ation of CO2 is around five times larger than produced by electron impact

excitation.

6.2.6. CS, S2

The other minor species whose production rates are of interest are the

sulphur containing molecules, CS and S2. The (0, 0) band of CS at 2576 A

has been observed in many comets mainly with the IUE satellite. It has also

been observed in the radio wavelength region. If the observed CS comes

from CS2, then CS2/H2O ≈ CS/H2O ≈ 0.4%. Therefore, CS2 is also a

trace constituent of the nucleus.

The molecule S2 was discovered in IRAS-Araki-Alcock from the obser-

vations of several bands of (B−X) system near 2900 A. The molecule S2 has

a short lifetime ∼ few hundred seconds at 1 AU. So it is present very close

to the nucleus. The derived production rate of S2 for the Comet IRAS-

Araki-Alcock observed at ∆ =0.032 AU is about 2× 1025/ sec. It has also

been seen in several other comets. The derived abundance ratio of S2/H2O

∼ 0.001.

6.2.7. Ions

The plasma tail is dominated by molecular ions of various kinds. The

most dominant among them are the ions CO+ and H2O+ whose lines lie in

the near blue and in the visible regions respectively. The early estimates

for the column densities of CH+ and CO+ ions were made based on the



observations of Comets Kohoutek and Bradfield for a projected distance of

about 104 km from the nucleus and for r = 0.5 AU. The deduced column

density (/cm2) of CH+, and CO+ for Comet Kohoutek are of the order of

1010.9±1.1 and 1012.6±0.9 respectively.

H2O+ was first identified in Comet Kohoutek in 1974. An analysis

of the surface brightness profiles of H2O+ in Comet Bennett and CO+

in Comet West has been investigated. The calculated production rate

of CO+ for r = 0.440 and 0.842 AU is Q(CO+) = 2.4 × 1028/ sec and

0.22 × 1028/ sec respectively with the heliocentric distance variation of

r−4.6±1.0. The estimated production rate of H2O+ for Comet Bennett is

about Q(H2O+) = 2× 1026/sec at r = 0.841 AU. This value may be com-

pared with the early estimates of Q(H2O+) = 3.7 × 1024/ sec for Comet

Kohoutek at r = 0.9 AU and Q(H2O+) = 2× 1023/sec for Comet Bradfield

at r = 0.6 AU.

The production rate of H2O+ can also be determined from the Doppler

shifted emission lines of H2O+ 6159 A and 6147 A emission doublets in the

(0, 8, 0) band based on the Fabry Perot technique. The observed ratio of

Q(H2O+)/Q(H2O) for Comet Halley is 1.2× 10−3 and for Comet Austin is

2.6× 10−3. The model based on magnetohydrodynamics and chemistry of

cometary coma indicate that for comets, upto 11% of water molecules are

finally ionized.

N+2

The reported identification of (B2Σ+u−X2Σ+

g ) system of N+2 ion near

3910A has been existing in the literature for the last few decades. The

presence of this band in Comet Halley was also reported in 1986. However

recent observation based on high spectral resolution and high signal to noise

ratio of several comets such as de Vico, Hale-Bopp and Ikeya-Zhang does

not seem to show the lines of N+2 at 3910 A. These observations have cast

doubt on the earlier detection of N+2 in comets, which was mostly based on

low-resolution spectra wherein the blending problem is severe arising out of

molecules CO+, CO+2 , CH and CH+. The derived upper limit from these

observations for N+2 /CO+ <∼ 10−5 − 10−4 which is a very low value. The

above study has great implication with regard to the presence of molecule

N2 in comets.

The molecule N2 is the least reactive among all species containing ni-

trogen. It is also believed to be the dominant equilibrium species in the

early solar nebula. Unfortunately, it is the most difficult molecule to make



any observation. It is difficult to observe from the ground because of the

presence of N2 in the earth’s atmosphere. The interpretation of the Giotto

Mass Spectral data is also difficult as N2 and CO have the same mass,

28. Therefore the presence of N2 in comets is inferred indirectly from the

presence of N+2 in comets. The method that is generally used to study N2

in cometary ices is through the study of its ion (N+2 (B −X)(0, 0) band at

3910 A). Hence N+2 has been used as a proxy for N2. However, non de-

tection of N+2 in some of the recent comets as discussed earlier and with

an upper limit to the abundance ratio of N2/CO<∼ 10−5 − 10−4 has raised

several important issues. Some of them are whether comets have varying

amount of N2 or did the depletion took place at a later time or is it related

to the place of formation in the solar nebula. May be the general view that

neutral N2 and CO should have been present in the early solar nebula is

questionable.

6.2.8. Complex molecules

Expected line intensities

The observations in the radio frequency region referring to source inten-

sity are generally expressed in terms of equivalent brightness temperature,

TB . This is related to the line optical depth τ through the relation, which

can be derived from the radiative transfer, as

TB = (1− e−τ )hν

k

[1

ehν/kTex − 1− 1

ehν/kTbg − 1

]. (6.44)

Here Tex is the excitation temperature, Tbg is the background brightness

temperature which can be taken to be 2.7 K and ν is the line frequency.

The optical depth at the line centre is given by

τ =c2

8πν2

2J + 3

2J + 1AJ+1→J

〈NJ〉∆ν

[1− exp(−hν/kTex)] (6.45)

where ∆ν is the line width in frequency units and 〈NJ〉 is the mean volume

density within the observed field in the rotational state J. For most of the

molecules, the lines are optically thin and therefore (1 − eτ ) ≈ τ in Eq.

(6.44). The average excitation temperature Tex can be calculated from the



relation

〈NJ+1〉〈NJ〉

=2J + 3

2J + 1exp

(− hν

kTex

). (6.46)

In order to proceed ahead, the excitation mechanism of the molecules

in a cometary atmosphere is required. To model the radio emission from a

molecule, it is necessary to consider the excitation scheme of the molecules

by taking into account the rotational, the vibrational and the electronic

states of the molecule. However, for most of the molecules in the coma,

the main excitation process is the pumping of the fundamental bands of

vibration by the solar infrared radiation field and the thermal excitation

by collisions in the inner coma. Such species do not have a significant

electronic excitation because they are generally predissociative and also

they lead to destruction rather than fluorescence. For the case of the stable

CO molecule the excitation rate of the ground state X1Σ+ vibrational levels

by solar infrared radiation field at 1 AU is about 2.6× 10−4/ sec for v′′ = 0

to 1 and 3 × 10−6/ sec for v′′ = 0 to 2. The rate is ∼ 10−6 to 10−9/ sec

for excitation from the ground level to higher electronic states. Hence,the

excitation rates are too low to populate electronic states as well as v′′ = 1

level. Therefore, it is a very good approximation to consider the excitation

starting from the v′′ = 0 ground vibrational state. Rotational excitation

by solar radiation is also negligible due to lesser fluxes at the wavelength

of rotational transitions.

To a very good approximation, the radiative excitation is dominated by

one or two vibrational bands. The balance between the infrared excitation

and spontaneous decay completely determines the rotational distribution

at fluorescence equilibrium. The fluorescence equilibrium can be evaluated

without much difficulty as it depends on the molecular constants and the

solar radiation field. However, the collisional excitation due to collisions

with H2O molecules is more complicated. One assumes spherical symme-

try and a reasonable value for collision cross-section between the molecule

and H2O and the coma temperature and velocity distribution derived from

hydrodynamic models. The molecular mean column densities can then be

derived by volume integration within the instrument beam. The excita-

tion temperature, Tex can therefore be estimated from Eq. (6.46). The

expected rotational line intensities can be calculated for reasonable values

of the relevant parameters in Eq. (6.44).

As an example the relative population distribution of linear molecules

CO and HCN is shown in Fig. 6.8. For such molecules, the rotational



Fig. 6.8 The rotational population distribution under resonance fluorescence equilib-

rium for CO and HCN is shown as a function of rotational quantum number and forlogarithmic ratio of the infrared excitation rate to the rate of rotational de-excitation

from the J = 1 level (Bockelee-Movan, D. and Crovisier, J. 1985. Astr. Ap., 151, 90).

population distribution is determined by the ratio of total infrared excita-

tion rate to the spontaneous decay rate of the rotational levels. Therefore,

the molecules with large rotational Einstein A coefficients like that of HCN

molecule, will have population concentrated mostly in the lowest rotational

levels as compared to molecules like CO and HC3N with small rotational

Einstein A values which have populations over a wider range of rotational

levels.

Observed molecules

As mentioned earlier, the line intensities in the radio region are generally

expressed in terms of equivalent brightness temperature TB . The brightness

temperature is related to the line flux Ful (upper level u, lower level l)

through Rayleigh–Jeans relation (i.e. hν <∼ kT ) of the Planck function.

The integrated line area over the velocity is related to the column density

through the relation ∫TBdv =

hc3Aul8πkν2

ul

〈Nu〉

where Nu is the column density in the upper level u and other symbols

have their usual meanings. In general, time-dependent excitation models



are required to derive the population distribution in the upper states for

calculating the total column density 〈N〉. Alternately from the observed

intensities of several rotational lines, it is possible to estimate the rotational

temperature which can then be used for the calculation of population distri-

bution in the upper states. Knowing the total column density of molecules

in the coma, the production rate of the molecule can be calculated. This

is then used for estimating the relative abundances of molecules in the

nucleus.

The number of known molecules in comets before Comet Halley’s ap-

parition in 1985-86 were limited in extent. However the concentrated effort

made on Comet Halley led to the detection of many new molecules. Sub-

sequently, large number of molecules of various kinds and complexities has

been detected in comets. Several lines arising out of isotopes have also

been detected. A typical observed spectrum in the radio region is shown in

Fig. 4.10. From such observations the production rate of molecules can be

derived. A wide variation in the observed abundance of parent species in

the coma of comets have been seen. In addition to the detected molecules

upper limits have been determined for many other molecules.

Several molecules belonging to CHO group, such as Methanol

(CH3OH), Formaldehyde (H2CO), Formic acid (HCOOH), Methyl Formate

(HCOOCH3), Acetaldehyde(CH3CHO) and Ethyl Glycol (HOCH2CH2OH)

have been seen. Surprisingly Ethyl Glycol is the most abundant organic

molecule in spite of its complexity and its identification is based on several

lines present in the millimetre spectra of Comet Hale-Bopp. The hydro-

carbons such as, Methane (CH4), Acetylene(C2H2) and Ethane (C2H6) are

present. Several sulphur bearing molecules like S2, H2S, OCS, H2CS etc

are present. Some of the complex molecules containing nitrogen are Methyl

Cyanide (HC3N), Isocyanic acid (HNCO) and Formic acid (NH2CHO).

Comet Tempel 1 showed the presence of PAHs.

The significant result that comes out of abundance studies of molecules

is the presence of chemical diversity among comets belonging to Oort cloud

and Jupiter-family of comets. For example, the abundance of CO varies

by a factor of around 40. The abundance of hydrocarbon CH4 varies by a

factor of about 10, while that of C2H6 shows less variation ∼ 0.4 to 0.7%.

The HCN abundance seems to peak around 0.1%. HNC, the isomeric form

of HCN was first detected (J(4-3) line at 363 GHz) in Comet Hyakutake and

subsequently in other comets. It is an unstable molecule under laboratory

conditions. The ratio of HNC/HCN in Comet Hale-Bopp is found to lie



in the range of 0.03 to 0.15. The abundance of H2S, H2CO and CH3OH

show large variations. In general, the abundance of most of the molecules

< 10−2−10−3 relative to water. Also the abundance of the molecules seems

to decrease with an increase in their complexity.

The observed diversity in the chemical composition among comets could

arise depending upon the local temperature and the nebular composition

in the region of their formation. The variation in chemical composition can

also arise if there was significant mixing of the nebular material. There-

fore it is not surprising that there exists a wide variation in the chemical

composition among comets.

The heliocentric variation of the production rate of molecules is of great

interest as it is directly related to the cometary ices, sublimation process etc.

However, monitoring along the cometary orbit is possible only for bright

comets. This was possible in the case of Comets Halley, Hyakutake, Hale–

Bopp etc. Since Comet Hale–Bopp was a bright comet and was discovered

when it was at 7 AU, it was possible to study the evolution of production

rate of a large number of molecules over a wide range of heliocentric dis-

tances. In Comet Hale–Bopp, CO was detected in the radio region up to

heliocentric distance ∼ 14 AU. CO is the main escaping gas at large helio-

centric distances as its sublimation temperature is very low ∼ 25 K. The

results derived from radio observations of molecules on Comet Hale–Bopp,

show that the chemical composition of the coma changed with heliocentric

distance. In addition, the heliocentric variation of the production rate of

molecules is different for different molecules.

Ethylene glycol is a known interstellar molecule and is observed towards

Sgr B2. The abundance ratio of ethylene glycol to CH3OH ∼ 0.1 in comets

whereas it is ∼ 0.001 towards Sgr B2.

Extended source of molecules

As discussed earlier, the results based on measurements carried out with

Neutral Mass Spectrometer on Giotto spacecraft of Comet Halley showed

different density distributions for the molecules H2O and CO in the coma.

The H2O distribution showed a R−2 variation consistent with simple radial

expansion from the nucleus, like a parent molecule, while the distribution of

CO required an additional extended source. Therefore the basic idea to look

for an extended source of any molecule is to examine the observed radial

distribution of the molecule in the cometary coma. For parent molecule

coming from or near the nucleus the radial distribution of the molecule

should fall steeply with radial distance, while for an extended source, the



variation initially is steeper and then flattens out. From the observed radial

distribution of molecules in the cometary coma of Comet Hale Bopp, the

extended source of H2CO, OCS and CO has been inferred. In contrast,

the observed column density profiles for the molecules H2O, C2H6, CH4

and HCN were found to be consistent with their release directly from the

nucleus.

6.2.9. O, C, N, S

Oxygen

The abundances or the production rates of oxygen atoms can be calcu-

lated from the observation of red doublet lines of oxygen atom occurring

at 6300 and 6364 A. Since the line at 6300 A is quite strong in comets,

it has been studied extensively for deriving the production rate of oxygen

atoms. Since these lines arise due to photodissociation of H2O, OH and CO

molecules which populate the 1D state (Fig. 4.11), the column densities of

the parent oxygen atoms can be easily obtained. The observed intensity of

the line is therefore given by

I ∝ α βN

τP(6.47)

where α and β are the yields for the particular state for dissociation and

branching ratio respectively. N is the column density of the parents and τPis the dissociation lifetime of the parent.The oxygen 1D production rate of

the comet can be calculated using the observed fluxes of the oxygen line

that has been corrected for various blending and other effects (Fcorr). The

photon rate is given by

P =Fcorr 4π∆2

(hν)6300. (6.48)

Here ∆ is the comet-earth distance. If the lifetime of the 1D state is τ

sec for the 6300 A line (∼ 100 sec), multiplying P by the lifetime will give

the total number of oxygen atoms within a cylindrical column of radius ρ,

M(ρ), centred on the nucleus, i.e.,

M(ρ) =Fcorr 4π∆2τ

(hν)6300atoms. (6.49)

The production rate is generally calculated using the Haser’s model

with H2O as the parent molecule and O as the daughter-product with

corresponding scale lengths lH2O and l0 respectively.



Fabry-Perot observations:

The main difficulty in the measurement of neutral oxygen from 6300 A

emission is the contamination due to NH2 emission lines. High resolution

spectroscopy allows the separation of spectrally NH2 lines from 6300 A line

thereby providing an uncontaminated 6300 A line for quantitative studies.

High resolution observations also help to avoid the airglow effects. The

Fabry-Perot spectrometer can be used for this purpose. An example of the

scan of 6300 A line is shown in Fig. 6.9. For the calculation of the produc-

Fig. 6.9 The observation of the cometary [OI] 6300 A line. The zero point for the ve-

locity scale is set at the position of the terrestrial airglow line. The solid curve representsa Gaussian least square fit to the data (Schultz, D., Li, G.S.H., Scherb, F. and Roesier,

F.L. 1992. Icarus 96, 190).

tion rate of O[1D] atoms, it is necessary to know the photodissociation of

molecules responsible for the production of O[1D] atoms. O[1D] may be

produced by the dissociation of H2O and OH through the reactions

H2O + hν → H2 + O[1D]

H2O + hν → H + OH

OH + hν → H + O[1D]



with respective branching ratios BR1, BR2 and BR3. O[1D] can also be

produced from the reaction

CO + hν → C + O[1D]

CO2 + hν → CO + O[1D]

with branching ratios BR4 and BR5. However, as CO and CO2 are mi-

nor constituents of comets and their photodissociation lifetimes are much

larger than those of H2O and OH, the contribution from these sources to

6300 A emission is relatively small within the field of view of observations.

Therefore, the contribution arising out of CO and CO2 is generally ne-

glected. The intensity of 6300 A emission at the projected distance ρ from

the nucleus is given by

I6300(ρ) =NH2O(ρ)

4πτ1+

NOH(ρ)

4πτ3. (6.50)

Here τ1 and τ3 are the photodissociation lifetime of H2O and OH leading

to O[1D] channels. NH2O and NOH are the column densities of H2O and

OH respectively at ρ. Using azimuthally symmetric parent-daughter Haser

model for representing NH2O and NOH, the intensity I6300(ρ) can be written

as

I6300(ρ) α1

ρ

∫ π/2

0

exp

(−ρ sec θ

l1

)dθ

+BR2.BR3

BR1 · ρ

[l1

l1 − l2

] ∫ π/2

0

exp

(−ρ sec θ

l1

)− exp

(ρ sec θ

l2

)dθ.

(6.51)

Here l1 and l2 are the Haser parent-daughter photodissociative scale length

for the (H2O, H) and (OH, H) respectively. By integrating Eq. (6.51) with

respect to ρ, it is possible to calculate the ratio of the total flux in the

6300 A line to the flux within the field of view of the observation. This

ratio is given by

AC =

∫ α0I6300(ρ)ρdρ∫ ρ1

0I6300(ρ)ρdρ

(6.52)

where ρ1 is the projected radius of the field of view of the observation.

Hence AC can be calculated, based on the values of the parameters and

theoretical branching ratios BR1, BR2 and BR3. The photodissociation



scale length can be derived from a comparison of Eq. (6.51) with several

high spectral resolution images of 6300 A emission. The production rate

Q(O1D) can be derived from the relation

Q(O1D) = (4

3)(4π∆2ΩI6300)AC. (6.53)

Here Ω is the solid angle of the field of view, I6300 is the average intensity of

the 6300 A emission within the field of view (in photons /cm2/ sec /ster).

The factor (4/3) takes into account the fact that 1/4 of the O1D atoms

radiate to the ground state via 6364 A channel.

To convert Q(O1D) to Q(H2O), the ratio of Q(H2O)/Q(O1D) has to

be known. This ratio can be determined from the photochemical branching

ratios. It can also be determined by comparing the production of Q(1D)

with the results of Q(H2O) derived from other observations. There is a

variation in the derived ratios of Q(1D) to Q(H2O). The estimated factor

for converting O[1D] production rate to Q(H2O) from the branching ratios

is given in Table 6.5.

Table 6.5 Branching ratios.

Reaction Theoretical values

H2O + hν → H2 + O(1D) 0.044, 0.054

H2O + hν → H + OH 0.840, 0.905OH + hν → H + O(1D) 0.103

Schultz, D., Li, G.S.H., Scherb, F. and Roesler,

F. L. 1993. Icarus 101, 95 and is for solar min-imum conditions. The individual references to

the above values are given in the paper.

It is around 11.7 for pre-perihelion observations and 13.7 for post-

perihelion observations for Comet Halley. The conversion factor for pre-and

post-perihelion observations are different as the branching ratio BR3 de-

pends upon the heliocentric velocity of OH. The resulting production rate

of H2O is compared with those derived from Hα observations in Table 6.6,

which shows a general agreement between the two determinations.

The [OI] 6300 A emission line profile from the coma of Comet Halley

was observed with Fabry - Perot during 1986. The observed profile is in

good agreement with the calculated profile based on Monte Carlo Particle

Trajectory Model. The derived water production rate is (2.90 ± 0.13) ×



1030 mol/sec. Fabry - Perot observations of [OI] 6300 A emission carried

out on Comet Hyakutake was also found to be in agreement with model

calculated line profile.

Table 6.6 H2O Production rate for Comet Halley from Hα and [OI]

6300 A observations.

Date of observation H2O Production rate (1029s−1)

Hα [OI] 6300 A

Dec 13, 1985 4.3±0.5 4.42 ±0.82

— 15, 1985 3.2±0.4 3.71 ±0.85

— 16, 1985 2.8±0.3 3.22± 0.61Jan 4, 1986 6.3± 0.8 7.65± 1.5

— 7, 1986 8.3± 0.9 9.49± 1.8

— 9, 1986 11.6± 1.5 12.1± 2.3— 12, 1986 7.8± 0.9 11.8± 2.3

— 13, 1986 17.5± 2.0 18.5± 2.6

(Adapted from Smyth, W. H., Marconi, M L, Scherb, F. and Roesler,F. 1993. Ap. J. 413, 756).

Green line of oxygen at 5577.3 A

The forbidden line of oxygen arising out of the 1S level which lies at

5577.3 A (1S - 1D) is highly blended with the Swan band sequence of

C2 corresponding to ∆v = −1. The study of this line can also help in

clarifying the problem of the parent molecule of oxygen atoms. The H2O

parent molecule model calculations have indicated the intensity of the green

line to be weaker by a factor of 10 or so compared to that of the red line.

Therefore, if the green line is detected, the intensity ratio of green to red

line could help in this regard. One way to extract the 5577.3 A line of

oxygen based on low dispersion spectra is through the comparative spectral

synthesis of ∆v = −1 C2 Swan band sequence. The calculated profile

which is a superposition of all the rotational lines arising out of bands

(0, 1) to (5, 6) and which has been corrected for the instrumental effect

show a slight excess of flux at λ ∼ 5577 A compared to the expected flux

that could be attributed to the contribution of the oxygen line of 5577 A.

These results indicate that the intensity of the oxygen line at 5577.3 A

should be roughly about 3 to 5% of the oxygen line at 6300 A. A similar

ratio was obtained from the measurements on Comet IRAS-Araki-Alcock

wherein the line 5577.3 A could be resolved. The line 5577.3 A has also been



resolved in Comets Halley, Hyakutake and LINEAR (C/1999S4). Since C2

is severely depleted in Comet LINEAR, 5577.3 A line could be seen without

any blending with C2 band. These results are, therefore, consistent with

H2O being the parent molecule for excited oxygen atoms.

The ultraviolet lines occurring at 1304 A have also been used to derive

the production rates of oxygen atoms (Table 6.7). The intercombination

doublet at 1356 A of OI has also been seen in some comets. Since the g

factor for this transition is too small, the excitation is likely to arise from

electron impact.

Table 6.7 Comparison of column densities of some species in comets.

N(O) N(C) N(S)

Comet Date r(au) ∆(au) (1013/cm2) (1012/cm2) (1012/cm2)

Bradfield 16 Jan 1980 0.80 0.40 2.5 1.7 1.7

31 Jan 1980 1.03 0.29 0.8 0.68 1.4Encke 5 Nov 1980 0.81 0.32 0.40 0.27 0.59

24 Oct 1980 1.01 0.29 − − −Tuttle 7 Dec 1980 1.02 0.50 1.2 1.0 1.3

(a) Weaver, H.A., Feldman, P.D., Festou, M.C., A’Hearn, M.F. and Keller, H.U. 1981.

Icarus 47, 449.

Carbon

The brightness of the carbon line at 1657 A arising out of the 3P state

has been used to derive the column densities of carbon for several comets

and are given in Table 6.7. The CI line at 1657 A was strong in the spectra

of Comet West. It could be accounted for from the dissociation of the

CO molecule. However in Comet Bradfield carbon emission is quite strong

inspite of CO being less abundant. This could indicate the presence of other

sources.

The detection of the carbon line at 1931 A arising from the metastable

level 1D state is of particular interest. This is because the parent molecule

should be able to explain the presence of carbon both in the 3P (1657 A

line) and 1D states.

There are several molecules which can give rise to carbon atoms such

as CO and several other minor species such as CO2 and so on. The in



situ measurements of Comet Halley showed that carbon is contained in

CHON grains. Therefore, this is also source of carbon atoms. In fact,

the inclusion of this carbon component from the grains to the observed

component from the fluorescent lines has removed the problem of carbon

deficiency in comets.

Nitrogen, Sulphur

It is not possible to make an estimate directly of the production rate

of nitrogen as its resonance spectral line of 1200 A has not been observed.

Therefore, the estimates have to be made indirectly, based on the nitrogen

bearing molecules like CN, NH, N+2 etc., and they give a rough value for

the ratio (N/O) ≈ 0.1.

The lines of SI 1812 A multiplet which was seen first in Comet West

has been seen in all the comets. The observed brightness of this multiplet

has been used to deduce the column densities for sulphur. The results for

some comets are given in Table 6.7.

Table 6.8 gives a rough representation of the elemental abundances of

H, C, N, O and S for comets.

Table 6.8 Average abundance ratios.

Ratios Comets Sun CI chondrites

H/O 2.0 1175 0.17

C/O 0.40 0.43 0.10N/O 0.09 0.13 0.007

S/O 0.03 0.02 0.07

Delsemme, A. H. 1991. In Comets in the

Post-Halley Era eds. R. L. Newburn, Jr.et al. Kluwer Academic Publishers, Vol.

1, p. 377.

The table shows that the observed ratio of C/O in comets is similar to

the solar value. However, the ratio H/O is very much smaller in comets

compared to the solar value. This implies that hydrogen is very much de-

pleted in comets. This in turn means that either the hydrogen has escaped

from the system or it was not trapped at the time of formation. A plau-

sible physical explanation could be that in the case of the sun it is the

gravitational binding while in comets it is the chemical binding which is

important. Hence, in the case of the Sun, gravity essentially prevents hy-



drogen from escaping, while in the case of comets only a small fraction of

hydrogen can be bound in molecules. Therefore, most of the hydrogen as

well as H2 disappear due to their high volatility. On the other hand, the

heavy elements like Na, Ca, Fe, Mn, etc are assumed to have sublimated

out of the refractory grains and have relative abundances similar to the

solar value (Fig. 5.19).

6.3. Analysis of Hydrogen Observations

6.3.1. Analysis of Lyman α measurements

The presence of a huge halo of hydrogen gas around comets was pre-

dicted by Biermann in 1968 based on the dissociation of the H2O molecule.

In general, the photon energy available for dissociation is much larger than

the dissociation energy. Therefore, this excess energy is usually carried

away by the dissociated products. For example, in the case of H2O the

excess energy gives a velocity ∼ 17.8 km/ sec to the hydrogen atom, and

this is much larger than the gaseous outflow velocity which is of the or-

der of 0.5 km/ sec. Since the lifetime for the ionization of the molecule is

∼ 106 sec, the observed extent of the size of the halo which is given by vτ

is about 106 to 107 km. Based on this simple physical argument and with a

hydrogen gas production rate of 1030 to 1031 molecule/sec, it was concluded

that the comets should be bright in the Lyman α line. The ultraviolet ob-

servations made in Lyman α with OAO-2 on Comets Tago-Sato-Kosaka

and Bennett confirmed the presence of the hydrogen halo extending up

to about 106 to 107 km around the nucleus. This result was further con-

firmed through a series of rocket and satellite observations made on various

other comets. In recent years, with the availability of better instruments

and spatial resolution, it has been possible to get good and high quality

isophotes of the Lyman α region. These high quality isophotes and line

profile have shown that they extend farther in the direction away from the

Sun compared to that in the direction toward the Sun. The isophotes also

become more and more elongated in the antisolar direction as the comet

goes nearer the Sun. These observations clearly show the effect of intense

Lyman α radiation pressure effects. The high quality isophotes also show

that the axis of the isophotes is not along the direction of the Sun-Comet

line, but is inclined at an angle to this line. The theoretical models have to

explain some of these observed effects.

In order to make a comparison with the observed isophotes, the emis-



sion of the hydrogen atoms has to be calculated. The collisions could be

important in the inner coma region < 104 km for the gas production rate

of about 1030 molecules/sec. However, the dissociation takes place at dis-

tances > 105 km and therefore the collisions are not important. For an

optically thin case the emission is given by

B = gN (photons/cm2/sec) (6.54)

where g is the emission rate factor and N is the column density of hydrogen

atoms. Therefore, the calculation of emission intensities essentially depends

upon N , which is model dependent.

To start with, one can assume an isotropic radial outflow without col-

lisions from a point source. The above assumption is reasonable since the

observed distribution of molecules is nearly spherically symmetric around

the nucleus. This is borne out by the shape of the isophotes in the visible

region. Therefore, the assumption of a point source with the radial outflow

of material usually termed as the Fountain model is reasonable for the in-

terpretation of the hydrogen coma. In this steady state model the hydrogen

atoms are released at the rate of QH from the nucleus and are pushed away

due to solar radiation pressure in the direction away from the Sun with a

lifetime tH .

The lifetime tH for hydrogen atoms is the sum total of lifetimes due to

photoionization (tph) and to charge exchange with protons (tpr). The cal-

culated photoionization lifetime at 1 AU is tph = 1.4×107 sec . The lifetime

for charge exchange with protons of flux Fw ≈ 2 × 108 proton/cm2/sec at

1 AU and for an effective cross-section σ = 2× 10−15 cm2 is equal to

tpr =1

σFw= 2.5× 106 sec .

Therefore, the total lifetime, tH is given by

tH =[t−1ph + t−1

pr

]−1

= 2.1× 106 sec . (6.55)

The acceleration due to the radiation pressure force is given by

b =hν

mHc

(πe2

mec

)fFr2. (6.56)

The velocity distribution of the hydrogen atoms is assumed to be

Maxwellian and is given by

f(v)dv =4√π

v2

v20

e−(v/v0)2 dv. (6.57)



The most probable speed v0 is related to the mean speed vH by the relation

vH =

(2√π

)v0. (6.58)

The density of hydrogen atoms at a distance r from the nucleus and for a

time of travel t is given by

nH =QH

4πvHr2e−t/tH . (6.59)

The distance r has now to be expressed in terms of the orbit of the hydrogen

atoms after they are released from the nucleus. The co-ordinate system is

chosen such that the nucleus is the origin, the Sun is in the negative z

direction and the positive x-axis is in the direction of the Earth. The

hydrogen atoms leaving the nucleus follow the parabolic orbits and they can

reach any point (x, y, z) with a velocity v through two different trajectories.

The total density which is made up of these two components is given by

the relation

nH(x, y, z) =QH4πv

[(a±√x2

0 − x2)√x2

0 − x2]−1

× exp

[− 1

tH

(2

b

)1/2√a± (x2

0 − x2)1/2

]. (6.60)

Here a = z + (v2/b) and x0 =√x2 − (z2 + y2). The parameter x0 is a

function of y and z and defines a surface of a rotational paraboloid about the

z-axis containing all the atoms within the maximum velocity v. The total

column density along the line of sight s, including the velocity distribution

of atoms, is given by

N(s) =

∫ α

0

v

vHf(v)2

∫ x0

0

nH(x, y, z)ds dv. (6.61)

The model-based isophotes of Lyman-α can, therefore be calculated for

different values of QH , tH and vH until a fit is obtained. The calculated

isophotes for Comet Bennett are shown in Fig. 6.10.



Fig. 6.10 Calculated Lyman α isophotes based on the Fountain and Syndyname modelsfor Comet Benett for April 1, 1970. The parameters used are: production rate QH =

1.2 × 1030/sec, outflow velocity νH = 7.9 km/sec, lifetime tH = 2.5 × 106sec at 1AU,

solary Lyman α flux = 3.2 × 1011 photons/cm2/sec/A (Adapted from Keller, H.U. 1976.Space Sci. Rev., 18, 641).

The number marked on each isophote gives the intensity expressed in

Kilo Rayleighs (KR) where 1 Rayleigh = 106 photons/cm2/ sec. The cal-

culated Lyman−α isophotes based on the above model were in qualitative

agreement with the observed isophotes; however in details, there were dif-

ferences. For example, the calculated isophotes were symmetrical about the

sun-comet line while the observed isophotes were inclined at an angle with

respect to this line. This reflects the limitations of the model. The main

assumption of the model is that the average parameters do not change with

time, and this implies a steady state. But this is far from true in a real

situation, wherein one has to consider the non-steady situations.

The non-steady state models were then developed and are usually re-

ferred to as Syndyname models. The formalism is very similar to the case of

the formation of the dust tail of comets, wherein one finds dust particles in

the tail emitted at different earlier epochs (Chap. 7). The hydrogen atoms

coming out of the nuclear region are being acted upon by the forces of radi-

ation pressure and gravitation. The resulting trajectories of the hydrogen

atoms in the orbital plane of the comet depend upon the ratio of these two

forces generally denoted as (1− µ). The particles ejected continuously and

having the same value of (1− µ), describe a curve called Syndyname. The

resultant effect of these two forces is to give curvature to the path of the

hydrogen atoms. The total column density of the hydrogen atoms is the

integral over all these paths corresponding to all the fictitious source points.



At the observation time t0, all the points are arranged along the Syndy-

name and correspond to the same emission time te. Let vP represent some

minimum velocity above which the contribution reaches the line-of-sight

integral. Then

vP =s

(t0 − te)where s represents the minimum distance from the fictitious source point.

The total contribution to the column density at time t is the integral over

all the surface densities and is given by

Nt(s) =1

2π

∫ α

vP

Pt(v) dv

v2t2[1− (vP /v)2]1/2(6.62)

where Pt(v)dv represents the production rate of hydrogen atoms with ve-

locities between v and v + dv. Here the subscript t is used to denote the

production rate as a function of time, which in turn depends on the helio-

centric distance. The total column density arising out of all the fictitious

source points on the Syndyname is given by

N =

∫Nt(s)dt =

1

2π

∫ ∞0

dt

∫ ∞vP

Pt(v)dv

v2t2[1− (vP /v)2]1/2. (6.63)

If the hydrogen atoms decay with a lifetime tH which may be a function

of t, then the exponential factor exp [−∫ t

0t−1H (t′) dt′] has to be included

in the expression for Nt(S). Therefore, the expression for the total column

density is given by

N =1

2π

∫ ∞0

dt exp

[−∫ t

0

t−1H (t′)dt′

] ∫ ∞vP

Pt(v)dv

v2t2[1− (vP /v)2]1/2. (6.64)

The velocity distribution is related to the production rate of hydrogen atoms

through the relation

Pt(v)dv =

(QHvH

)vf(v)dv. (6.65)

Figure 6.10 shows the calculated isophotes for Comet Bennett based on

the above model for the same parameters of vH , tH , QH and F used in

the Fountain model. The figure clearly shows that the isophotes are now

inclined at a certain angle to the Sun-Comet line and are symmetrical about

the Syndyname, and this is consistent with the observations.

The model described above can be improved further by assuming the

parent molecules to be released by the nucleus and allowing for the decay

of the molecules as they flow outwards. This eliminates the assumption of



the hydrogen atoms being emitted from a point source. In this model the

parent molecules are released isotropically from the nucleus. These parent

molecules could give rise to two or more decays along their paths with a

certain value for the scale length for each decay. This in turn is dependent

on the lifetime and the velocity of the molecules. The formalism developed

in Sec. 6.1.2 can easily be applied to the present situation.

The models described above has been applied to the Lyman−α obser-

vations of many comets with great success. A comparison of the observed

and the calculated isophotes allows one to determine quantitatively the

production rate, the lifetime and the outflow velocity of hydrogen atoms.

The detailed observations of Comet Bennett showed that two Maxwellian

velocity distributions in the model calculation corresponding to vH = 7 and

21 km/sec with 50:50 of each are required to fit the observations.

Monte-Carlo approach

In the formulation outlined above many details are still to be under-

stood. This, therefore, led to the consideration of Monte-Carlo approach.

The model developed is a three-dimensional time dependent Monte-Carlo

particle trajectory model (MCPTM). It takes into account the physical

processes which are important in the inner and outer coma of comets.

In particular, it includes a physically realistic description of the detailed

production mechanism and trajectories of H atoms produced by the pho-

todissociation of H2O and OH. It takes into account the solar radiation

pressure acceleration. The gas dynamic model calculates the velocity and

temperature variation in the coma and includes the multiple collisions, pho-

tochemical heating, radiative transfer and so on. The model couples the

simple gas-dynamic flow of the gas with the MCPTM formalism. There-

fore, the gas-dynamic model coupled with MCPTM formalism considers in

a consistent way and also in a realistic manner the actual trajectories of

as many as ∼ 105 radicals or atoms. The space and column densities and

hence the emission rates are then calculated. For comparing with the ob-

servation, the calculated results have to be referred to the projection on the

sky-plane which in turn depend upon the relative geometry of the Earth,

the Sun and the comet at the time of the observations. These are then

compared with observations.

The model simulated isophotes based on MCPTM formalism is in

agreement with the observed isophotes of Comet Hale-Bopp (Fig. 6.11).

Therefore MCPTM formalism which takes into account in detail the im-

portant physical processes in the inner and outer extended coma region, is



Fig. 6.11 (c) Comparison of the SOHO SWAN observations of Ly α isophote contours

(solid lines) of Comet 1995 O1 (Hale-Bopp) at r = 1.75 AU with the calculated isophotesbased on MCPTM models. (b) Shows the H-atom velocity distribution upon photochem-

ical production (thin line) and after Monte Carlo model calculation (thick line) (Combi,

M.R. et al. 2000. Icarus, 144, 191).

a very general method of studying the observations of Lyman α in comets.

The additional complications in terms of physics and dynamics can be in-

corporated as and when the need arises.

6.3.2. Analysis of Hα observations

The Hα line at 6563 A can also be used to study the daughter species and

could in turn be used to monitor the production rate of H2O. Unfortunately,

this line is usually faint and in addition could be contaminated with geocoro-

nal Hα emission, diffuse galactic Hα emission particularly when the comet

is located near the galactic plane and cometary H2O+ (0, 7, 0) emission at



6562.8 A. However, the observations carried out with the Fabry-Perot spec-

trometer, the Doppler shift of the cometary Hα may be sufficient to avoid

the geocoronal Hα emission (Fig. 6.12). The contamination of galactic Hα

Fig. 6.12 Fabry-Perot observation of Hα(6563 A) line of Comet Halley taken on January

4, 1986. The smaller feature which is centered at zero velocity is the cometary Hαemission while the larger feature centered around - 32 km/sec is the geocorona Hα

emission. The model fit to the observed cometary Hα is shown by the dashed line(Smyth, W.H., Marconi, M.L., Seherb, F. and Roesier, F. 1993. Ap. J., 413, 756).

emission and H2O+ emission can be minimized by taking proper precau-

tions in making the observations. The commonly used Haser’s model is

not adequate for the analysis of Hα emission as the hydrogen atoms have

large dispersion in velocities. These had limited the study of Hα emission

in comets.

As discussed earlier, the Monte-Carlo Particle Trajectory Model

(MCPTM) has been successfully applied to Lyman α observations of

comets. This formalism can be modified and used for the analysis of ob-

servations of Hα emission in comets. The main quantity to be determined

in the application of MCPTM model is the g-factor for the Hα emission.

This has to be calculated taking into account the contributions from all the

transitions which can give rise to this emission. This involves the knowl-

edge of the transition probabilities of all the relevant transitions which



contribute to this emission and the solar radiation which excites these lev-

els i.e., through excitation and de-excitation. The principal source of Hα

emission is through the excitation of H atoms by solar Lyman β pho-

tons. The actual profile of Lyman β has to be used since H atoms are in

motion. For solar minimum conditions the total flux of Lyman β line at

1AU is 3.5×109 photons/cm2/ sec while the corresponding Lyman α flux is

3.1× 1011photons/cm2/ sec . For an accurate calculation of g-facotr for Hα

emission, in addition to principal contribution from the 3P to 3S transi-

tions, several other secondary contributions have also been considered. The

derived g-factor is velocity dependent, as in the case of OH (see Fig. 5.14).

The MCPTM can be used to simulate observations of Hα emission.The cal-

culated and the observed line profile for Comet Halley is shown in Fig. 6.12.

The derived velocities for such a fit require a broad velocity peak from 5 to

9 km/sec representing the bulk of (∼ 70%) of H atoms released and a veloc-

ity ∼ 8 km/sec due to the dissociation of OH. Additional smaller peaks also

appear at around 11 km/sec (∼ 7%) and 22−26 km/sec (∼ 20%) due to the

formation of H from OH and at 15 to 19 km/sec due to photodissociation of

H2O. A comparison of the derived production rate of H2O from Hα emis-

sion with those derived from 6300 A emission observations should help in

checking the photochemical processes which give rise to atomic oxygen and

hydrogen from the parent molecule H2O. Fortunately such a comparison

can be made for Comet Halley, as measurements of Hα and 6300 A emis-

sions were carried out on the same dates, with the same instrument, same

calibration procedure and so on. The results derived from such an analysis

are shown in Table 6.6. The agreement is reasonable in view of the various

uncertainties involved in the calculation of parameters such as, g-factor for

Hα emission and the branching ratios for the production of O[1D] atoms

from photodissociation of H2O and OH.

6.4. Related Studies

6.4.1. Gas-phase chemistry in the coma

The understanding of the formation of various observed species in the

coma has attracted much attention in recent years. They could be produced

either through the break-up of complex molecules or by some other physical

process, as the gas expands outwards in the coma. Therefore, the distri-

bution of the observed species in the coma does not necessarily reflect the

original gaseous material ejected from the nucleus. In addition to the var-



ious physical processes that could modify this gas, the gas-phase chemical

reactions among various constituents (neutral and ions) in the inner coma

could also alter the original gas. The chemical reactions can take place

due to the fact that the timescales involved for many of the reactions are

much faster than the solar photodestruction rates at a heliocentric distance

of 1 AU. In fact, the early investigations did indicate that ion-molecule

could be quite important and various reaction sequences were considered.

They did produce substantial amounts of various types of species that have

been observed in comets. However, the reaction schemes considered were

quite simple so that the calculations could be carried out without much

difficulty. But in a real physical situation various networks of chemical re-

actions have to be considered between neutrals and ions of different atoms

and molecules.

The nucleus is assumed to be spherical in shape containing the volatile

material in certain proportions. The gases evaporate from the nucleus as a

result of the solar heating. The vapourization theory (Chap. 11) could be

used to calculate the gas production rate, outstream velocity, temperature,

etc., which is consistent with the assumed mixture. The total number

density present in the expanding coma gas at a distance r from the nucleus is

related to the density at the nucleus surface n(R0) through the conservation

of mass by

n(r) = n(R0)

(R0

r

)2

. (6.66)

Here R0 is the radius of the nucleus. To take into account the various

physical processes and the chemical reactions which modify the particular

species, it is necessary to have the particle conservation equation. The

equation for a general case for a species i can be written in the form

∂ni∂t

+∇ · nivi =∑i

Rji − ni∑k

Rik, (6.67)

where ni and vi denote the number density and the flow velocity respec-

tively. The first and the second terms on the right-hand side of the equation

denote the rate of formation of species i, by processes involving species j

and the rate of destruction of species i through processes involving species

k. The whole set of equations given by Eq. (6.67) are coupled to each

other through sources and sinks which are on the right-hand side of the

equation. The assumption of spherical symmetry and a constant outflow

velocity for all the species will simplify the equations considerably. Pho-

todissociation and photoionization processes as well as the opacity effects



for the incident solar radiation are taken into account. The various chem-

ical reactions between ions, electrons, neutrals and free radicals have been

considered (Table 6.9).

Table 6.9 Gas phase chemical reactions with examples.

Photodissociation hν + H2O→ H + OHPhotoionization hν + CO→ CO+ + e

Photodissociative ionization hν + CO2 → O + CO+ + eElectron impact dissociation e+ N2 → N + N + e

Electron impact ionization e+ CO→ CO+ + e+ e

Electron impact dissociative ionization e+ CO2 → O + CO+ + e+ ePositive ion-atom interchange CO+ + H2O→ HCO+ + OH

Positive ion charge transfer CO+ + H2O→ H2O+ + CO

Electron dissociative recombination C2H+ + e→ C2 + H

Three-body positive ion-neutral association C2H+2 + H2 +M → C2H+

4 +M

Neutral rearrangement N + CH→ CN + HThree-body neutral recombination C2H2 + H +M → C2H3 +M

Radiative electronic state deexcitation O(1D)→ O(3P ) + hν

Radiative recombination e+ H+ → H + hνRadiation stabilized positive ion-neutral association C+ + H→ CH+ + hν

Radiation stabilized neutral recombination C + C→ C2 + hνNeutral-neutral associative ionization CH + O→ HCO+ + e

Neutral impact electronic state quenching O(1D) + CO2 → O(3P ) + CO2

Electron impact electronic state excitation CO(1Σ) + e→ CO(1Π) + e

(Huebner, W.F., Boice, D.C., Schmidt, H.U. and Wegmann, R. 1991. In Comets in the

Post-Halley Era, eds. R.L. Newburn, Jr. et al., Kluwer Academic Publishers, p. 907).

A step-by-step solution of the coupled Eq. (6.67) gives the number den-

sity of the species and its variation with r. Figures 6.13 and 6.14 show

the typical results of such calculations for various neutrals and ions. The

column densities can be calculated by integrating the number density along

the line of sight.

It is interesting to note from the model calculations (Fig. 6.14) that

the most abundant ions in the inner coma should be H3O+, NH+4 and

others. Therefore, in a comet, the most abundant ion in the inner coma

should be H3O+. The dominant H3O+ in the inner coma arises as a result

of the production of H3O+ through photoionization of H2O followed by



Fig. 6.13 Typical variation of the number density of various species plotted as a function

of the distance from the centre of the nucleus. The radius of the nucleus = 1 km andr = 1 AU (Huebner, W.F. and Giguere, P.T. 1980. Ap. J., 238, 753).

ion-neutral reaction and destruction by dissociative recombinations,

i.e., H2O→ H2O+ + e,

H2O+ + H2O→ H3O+ + OH

and H3O+ + e→ H2O + H.

Such calculations have been used to identify the species found by Giotto

Mass Spectrometer.

6.4.2. In situ mass spectrometer for ions

The in situ measurements carried out with spacecrafts on Comet Halley

gave confirming evidence for some of the basic ideas of gas-phase chemistry

discussed earlier. In fact the gas-phase chemistry has been used extensively

in the interpretation of ions observed in the inner coma of Comet Halley to

extract the production rates of ions.

The ion mass spectrometer on board the Giotto spacecraft has pro-

vided important information about the ions present in the coma of Comet

Halley to a distance ∼ 1000 km. The mass resolution is excellent. This

makes it easier to calculate the ion densities. The instrument measures

the distribution of m/q. Therefore, a careful analysis of the data with the



Fig. 6.14 Typical variation of number density of ions plotted as a function of the dis-tance from the centre of the nucleus. (Huebner, W.F. and Giguere, P.T., 1980. op.

cit).

theoretical modelling is required before the peaks can be identified with

certainty. However, the singly ionized peaks can easily be identified. The

peaks at m/q = 18, 17 and 16 gave the direct detection of H2O as these

peaks belong to the H2O group of ions, i.e., H2O+, OH+ and O+. The peak

at m/q = 12 is certainly C+. The peak at m/q = 19 is almost certainly

H3O+ which supported the chemical models. Molecular ions have been

detected ranging beyond 100 amu. The major ion species and also some

of the secondary ones which can contribute to the observed m/q peaks are

shown in Table 6.10.

The important reactions for forming some of these ions are shown in

Fig. 6.15. As can be seen from Table 6.10, NH+4 can also make contribution

to the peak at q/m = 18. Similarly, NH+3 could contribute to the peak at

q/m = 17. Several peaks have been seen with the positive ion cluster

analyzer on Giotto spacecraft with an alternate mass difference of 14 and

16amu as can be seen from Fig. 6.16. A suggestion has been made that

it could arise due to ion fragments CH2 and O breaking off from the large

parent molecule formaldehyde (H2CO)n, also known as Polyoxymethylene

(POM).



Table 6.10 Dominant ion species contributing to mass range detected in

the ion mass spectrometer.

M/q Ion Species M/q Ion Species

12 C+ 34 34S+, H2S+

13 CH+, 13C+ 35 H3S+

14 CH+2 , N+ 36 C+

3

15 CH+3 , NH+ 37 C3H+, H3O+.H2O

16 O+, CH+4 , NH+

2 38 C2N+, C3H+2

17 OH+, NH+3 , CH+

5 39 C3H+3

18 H2O+, NH+4 40 C3H+

4 , CH2CN+

19 H3O+ 41 C3H+5

20 H182 O+ 42 -

21 H183 O+ 43 CH3CO+

22 - 44 CO+2 , CS+, C3H+

8

23 - 45 HCS+, HCO+2

24 C+2 46 H2CO+

2 , NS+, H2CS+

25 C2H+ 47 H3CS+, HNS+

26 C2H+2 , CN+ 48 SO+

27 C2H+3 , HCN+ 49 HSO+

28 H2CN+, CO+, N+2 , C2H+

4 50 C4H+2

29 HCO+, C2H+5 , N2H+ 51 C4H+

3

30 H2CO+, CH4N+, NO+, C2H+6 52 C3H2N+

31 CH2OH+, HNO+ 53 C4H+5

32 S+, O+2 54 -

33 HS+, O2H+ 55 C3H3O+, C4H+7

(Huebner, W.F., Boice, D.C., Schmidt, H.U. and Wegmann, R. 1991. In

Comets in the Post-Halley Era, eds. R.L. Newburn, Jr. et al., Kluwer

Academic Publishers, p. 907).

6.4.3. Temperature and velocity of the coma gas

The knowledge of the temperature and velocity of the coma gas as a

function of distance from the nucleus is required for the calculation of the

excitation process as well as for the derivation of molecular abundances.

This can, in principle, be derived from the model calculations which take

into account the thermodynamical and hydrodynamical processes in the

coma. For the description of the state of the gas, the equations of mass,

momentum and energy have to be considered. The corresponding fluid

dynamic equations with velocity u, mass density ρ and pressure P can be

written as∂ρ

∂t+∇ · (ρu) = ρ, (6.68)

∂

∂t(ρu) + (u · ∇)ρu+ ρu(∇ · u) +∇P = q, (6.69)



Fig. 6.15 For illustrative purpose, this figure shows the network of major chemical reac-

tions involving the formation and destructive processes that have to be considered in thecalculation of abundances of some of the species. The arrow indicates the direction of the

reaction ν represents photodissociation or photoionization and e indicates electron dis-

sociative recombination (Huebner, W.F., Boice, D.C., Schmidt, H.U. and Wegmann, R.1991. In Comets in the Post-Halley Era,eds R.L. Newburn, Jr. et al., Kluwer Academic

Publishers, p. 907).

∂

∂t(ρu2

2+ ρε) +∇ ·

[ρu(

u2

2+ h)

]= E. (6.70)

Here ε and h are the specific internal energy and enthalpy respectively.

The quantities on the right hand side represent the sources and sinks for

the mass, momentum and energy respectively. These equations have to

be solved with the appropriate initial boundary conditions. The equations

are generally solved in a one-dimensional form with radial variation. Such



Fig. 6.16 The ion mass spectrum of Comet Halley observed on March 14, 1986 with the

Giotto spacecraft at a distance 11 -12 × 103 km from the nucleus. The peaks at regular

intervals with alternate difference in mass number of 14 and 16 have been interpreted asdissociation production of the molecule polyoxymethylene. (Huebner, W.F. et al. 1987.

Symposium on the diversity and similarity of comets, ESA-SP 278, p. 163; see also

Vanysek, V and Wickramasinghe, N.C. 1975. Astrophys. Space Sci., 33, L19).

solutions are sufficient in the inner coma, but in the outer coma where

the solar wind interactions occur, the description of the state of the ionic

gas may require two or three-dimensional solutions. The corresponding

equations for a spherically symmetric coma for the one dimensional case

are given by

1

r2

d

dr(r2uρ) = ρ, (6.71)

ρudu

dr+dP

dr= q (6.72)

and

1

r2

d

dr

[r2ρu(h+

u2

2)

]+

[γ

γ − 1ρu

]= Q− L. (6.73)

Here γ is the specific heat ratio for the gas. Q and L are the energy input

and energy loss rates respectively. The above equations can easily be mod-

ified to include the dust component. The method is to write the analogous



equations for the interaction of dust and the gas which are therefore treated

as separate fluids.

The results of the solution of the equations depend on the various phys-

ical processes that have been considered in the terms on the right hand side

of the equations. In particular, it depends on the source of energy input,

energy loss, chemistry and so on. Therefore, the emphasis on the solution

depends upon the problem of interest. For example, several studies have

been carried out which mainly emphasize certain aspects like nucleus-coma

interface, gas-dust interaction, detailed study of the chemistry or the so-

lar wind interaction and so on. Here again initially simple models were

considered.

The inner part of the coma is the collision dominated region (LTE).

It decreases in importance moving away from the nucleus. The hydrody-

namic flow which is a good approximation in the inner region becomes free

molecular flow in the outer region of the coma, with a transition region

in between. Therefore, the solutions of the hydrodynamic equations can

represent approximately the real solution provided proper quantities are

used referring either to LTE or NLTE situations and a suitable form for

in-between regions. The photolytic heating is given by the thermalisation

of the excess kinetic energy carried off by the photolytic products due to

photodissociation or exothermic chemical reactions. The dominant contri-

bution to the heating process comes from the dissociation of H2O through

the channel H2O+hν →H + OH + Energy. In this process H atoms has

an excess of velocity ∼ 17.5 km/sec and OH radical ∼ 1.05 km/sec. This

means that collisions have to be important which imply that the time scale

for collision must be smaller than the time scale for the expansion of the

gas.

The cooling process of the coma gas comes mainly due to H2O molecules

because of its large abundance. The cooling takes place mainly through

their strong rotational transitions.

In a real cometary situation, the calculation of photochemical heating

is not simple as the coma gas is not in LTE. Also the coma region where

the density is high for the local photochemical effect to be highly efficient,

depends upon the gas production rate and the expansion velocity in the

coma. Since the gas density varies as 1/r2, the photochemical heating

efficiency decreases slowly and hence it is rather difficult to define a collision

zone for heating caused by super-thermal H atoms.

Some of the complications in the calculation of cooling rates have also to

be considered. In particular, the water density in the coma is high enough



that some of the lines could be optically thick. The optical depth effect is

generally calculated based on escape probability method. The net result

being that rotational temperature basically comes out of equilibrium with

the gas kinetic temperature. Several of these effects and refinements have

been incorporated in the calculation of heating and cooling rates which

greatly influence the temperature and velocity of the gas in the coma.

The results of such model calculations provide the velocity and tem-

perature variation in the coma. The resulting temperature distribution is

shown in Fig. 6.17. The temperature distribution falls steeply to a mini-

mum temperature ∼ 20 K at a distance ∼ 102 km and then increases again.

The general shape of the temperature variation is the resultant effect of the

relative importance of heating and cooling rates of H2O and the expansion

cooling.

Fig. 6.17 The calculated variation of gas temperature as a function of distance fromthe nucleus for Comet Halley. (Combi, M.R. 1989. Icarus, 81, 41).

Refined models

All the discussion presented so far pertaining to chemistry, temperature

and velocity of the coma gas were based on certain simplifying assumptions

in the model calculations. In addition several processes that can occur

in the coma were considered individually or in combination with others.

But in a real situation, due to the nature of the physical conditions in the

coma, various processes operate simultaneously in a complex and intricate

manner.



The starting point for all the activities seen in the coma of a comet is

due the absorption of solar radiation by various molecules. This leads to

photodisssociation and ionization of the parent molecules. These in turn

fragment giving rise to newer species and the process continues. The energy

budget in this process is controlled by the heating and cooling mechanisms

which in turn has a great effect on the coma gas. The earlier assumption

of considering the gaseous coma as a fluid is reasonable in the inner parts

of the coma where the densities are high. But moving outwards from the

nucleus to the coma, the density gradually decreases and the flow of the gas

becomes more like a free molecular flow. In these regions the description

of the gas as a fluid is not applicable. Therefore it is necessary to consider

the flow of the gas in the coma in a consistent manner going from a region

of higher density to regions of decreasing densities. The presence of dust

can also have an effect on the structure of the coma. This comes through

the change in the temperature of the dust particles, which in turn can have

an effect on the vapourization of volatiles from the CHON dust particles.

Therefore the observed structure of the coma gas, is a resultant effect of

coma chemistry, physics and dynamics that are intimately connected with

each other. Therefore all these effects have to be considered simultaneously

in any realistic model.

Several types of such models have been developed. Some of them are,

Hydrodynamic model, Monte Carlo Model. Hydrodynamic model has been

used extensively for the study of the coma gas. For example, a chemi-

cal/dynamical model developed for the study of the coma includes in the

chemical reaction scheme involving around 180 species and more than 3500

reactions.

The Monte Carlo approach is the most general and most accurate de-

scription of the coma which can take care of all the complexities involved

within the coma. One such model is called Direct Simulation Monte Carlo

Method (DSMC). This is a time dependent in which the molecular motions

of a large number of molecules are followed and simultaneously their re-

sponse to imposed field. This model has been used widely in the study of

the coma gas and has been highly successful. As an example, the observed

gas velocity variation in the coma of Comet Halley deduced from Giotto

observations is compared with the results based on time dependent DSMC

model in Fig. 6.18 for distances between 103 and 3× 104 km. The observed

velocities are deduced from the energy shifts of peaks in the neutral mass

spectrometer energy spectra and show a variation from a value of about 0.8

to 1.05 km/sec. As can be seen the observed and the model based velocity



distributions are in reasonable agreement.

Fig. 6.18 The calculated variation of expansion velocity with distance from the nucleus

based on time-dependent DSMC model is compared with the observed variation for

Comet Halley deduced from the Giotto measurements (Combi., M.R. 1996. Icarus, 123,207.

6.5. Parent Molecules

Most of the observed species in comets cannot exist as such in the nu-

cleus. Therefore, it is suggested that they could arise out of the photodisso-

ciation process of stable complex molecules. However, it is very difficult to

observe these parent molecules directly from the Earth-based observations

due to their short lifetime, as they are dissociated in most of the cases very

near the nucleus. Hence the information about the possible parent molecule

has to come from indirect means and, in particular, mainly from the study

of the decay products. However, the spacecrafts which passed by Comets

Halley, Tempel 1 and Wild 2 made in situ measurements of the inner coma



and have given important and direct information about the possible nature

of the parent molecules. In general, there are several ways to approach this

problem.

The straightforward approach is to look for the dissociated products

of simple parent molecules as in the case of H2O molecules. There are

not many clear-cut cases of this type. The laboratory investigations of

the photochemistry of various types of molecules can help a great deal in

elucidating the problem of the parent molecules. The laboratory studies

have to give information with regard to the products of photodissociation,

the energy required to initiate the reaction and also the branching ratios

for various paths of the reactions. At present, the data is limited in extent.

Also such studies have to be performed for all molecules of astrophysical

interest. To start with, the various types of molecules identified in the

interstellar space could be used. The calculated photochemical lifetimes of

several molecules based on the laboratory data show that for most molecules

the lifetime is below 2 × 104 sec at 1 AU. The results based on laboratory

photochemical studies should be used only as a guideline when applying

the same to a cometary atmosphere. This is due to the fact that the

astrophysical situations are vastly different from laboratory conditions.

The surface brightness distribution as a function of the radial dis-

tance from the nucleus can give information about the parent molecule

(Sec. 6.1.2). The comparison of the observed brightness profiles with those

of calculated profiles gives the scale lengths of the species. The lifetime

of the species, can therefore, be obtained by dividing the scale length by

the expansion velocity. These lifetimes in turn can give information with

regard to the possible types of the parent molecule.

Another approach is through the study of gas-phase reactions in the

coma. If an agreement can be obtained between the observed and the

calculated abundances of various species, one can hopefully trace backwards

through the network of reactions and this could help in identifying the

plausible parent molecule of various species. However, the procedure may

not be very practical in most of the cases.

Another possible method is through the spectral study of the observed

lines and particularly from those molecules which give lines arising out of

singlet-singlet and triplet-triplet transitions. The idea is to see whether the

observed intensities of lines arise preferentially from the singlet-singlet or

the triplet-triplet states of the molecule. This could in turn be related to

the products arising out of the dissociation of the parent molecule.

The study of forbidden lines can also help in identifying the parent



molecule of the species which gives rise to these lines. By combining differ-

ent types of studies it is possible to infer the possible nature of the parent

molecule of some of the observed species. We may illustrate them with a

few examples.

Until the direct detection of H2O from Comet Halley in 1986, all the

earlier observations had built up a strong case for the H2O as the dominant

volatile component of the nuclei of comets. This is basically related to the

fact that all the dissociated products of H2O are seen and they dominate

the ultraviolet spectra of comets. But the direct detection of H2O came

from the observation made from the Kuiper Airborne Observatory of the ν3

band near λ ∼ 2.65 µm in Comet Halley. The same band was also observed

by the infrared spectrometer on the Vega spacecraft. The results based on

Neutral Mass Spectrometer (NMS) measurements on the Giotto spacecraft

showed that H2O comprised around 80% of the volatile of Comet Halley.

The presence of CO in comets was indicated by the detection of fourth

positive bands at λ ∼ 1500 A. Its abundance relative to water varies from

a few percent to around 30%. For Comet Halley, the value derived from

various observations is ∼ 17%, while NMS data gave a value ∼ 7%. The

lower value is attributed to CO coming from the nucleus and the rest is

believed to arise from an extended source. CO is the second most abundant

molecule in comets.

The observation of the ν3 band of CH4 around 3.3 µm has yielded an

estimate for the production rate of CH4 which is ∼ 1%.

Although CO2 was believed to exist due to the presence of CO+2 ion in

comets, it was first detected in Comet Halley from the observation of ν3

band at 4.3 µm from the IKS instrument in the Vega spacecraft and the

NMS instrument on Giotto which gave a value ∼ 3% relative to water. The

observed distribution of the column density of CO2 is in good agreement

with the ρ−1 law. This is consistent with the distribution of R−2 expected

for parent molecules in the vicinity of the nucleus.

NH3,NH2 and NH have been seen in comets. The photodisssociation of

NH3 appears to produce mostly NH2(∼ 95%) and very little NH(∼ 5%).

Therefore the parent of NH2 is NH3 and the dominant parent of NH is

NH2.

The photodissociation of HCN, namely

HCN + hν → CN + H

can explain the CN radical both from the point of view of production rates

as well as from the kinematic considerations. The molecule C2N2 has also

been proposed as the parent of CN.



The parent of CH is most like CH4. Since CH3 is highly unstable, CH4

first decays to CH2 which in turn decays to CH.

i.e., CH4 → CH2 + H2,

CH2 → CH + H

The source of S2 (lifetime ∼ 350 sec) has been the subject of considerable

debate as the presence of S2 in the cometary nuclei has great significance

for the origin of cometary material. CS2 hardly photodissociates into S2.

It is highly unlikely that solid S2 has formed from gas phase condensation,

which primarily produces other compounds. So is also with the irradiation

of the nucleus by cosmic rays or ultraviolet radiation and so on.

The long-slit spectroscopic observations has been carried out with HST

to get the spatial brightness profile of S2 in Comet Ikeya-Zhang. These ob-

servations seems to indicate that most likely S2 originate from the nucleus.

Therefore, S2 could be a parent molecule.

The most plausible parent of CS is the short-lived (life time at 1 AU ∼500 sec) CS2 molecule,

i.e., CS2 + hν → CS + S(3P ),

CS2 + hν → CS + S(1D).

Another possible source of CS is OCS. However the main dissociation pro-

cess of OCS is CO and S. Hence the contribution of OCS to S is minor.

The S produced from CS2 and S2 does not appear to account for the

observed amount of sulphur. The molecule H2S which has been detected

in comets may contribute to the observed sulphur.

The analysis of the C2 brightness profile seems to suggest that it could be

formed in a three-step photodissociation process of some parent molecule.

The laboratory studies indicate that C2 can be produced in a two-step

process as follows:

C2H2 + hν → C2H + H

followed by

C2H + hν → C2 + H.

The C3 molecule is a difficult case for which there are some suggestions.

Laboratory studies have shown that C3 molecule can be formed from C3H4.

This molecule photodecomposes to give C3H3 and C3H2 as intermediate

products, which can then absorb a photon to give C3 molecule. C3H8 has

also been suggested.



Table 6.11 gives a limited list of probable parent molecules of some

observed species in comets. Some of the identifications are to be taken

with caution as they are still preliminary in nature.

Table 6.11 Possible parent molecules and abun-

dances of some of them.

Observed species Possible parent molecule

H2O, H2O+, OH, H, O H2O

CO+2 , CO+, CO CO2

CO, CO+, C CO

CH CH4

CN HCN, C2N2

HCN HCN

NH, NH2 NH3

NH NH2

C2 C2H2, C3H2O

C3 C3H4, C3H8

CS CS2, OCS

S2 S2

S S2, OCS, CS2, H2S, SO2

6.6. Chemical Diversity

The surprising result that comes out of the study of the abundances of

a large number of molecular species in comets, is the presence of chemical

diversity among them. This could help in the understanding of the possible

sites of formation of these objects as well as the physical processes that

could have taken place leading to such an observed effect.

The other problem of great interest is the bulk composition inside the

nucleus of a comet. The only available information for the study is the pro-

duction rate of volatiles and its variation with heliocentric distance. The

natural question that arises is, to what extent the observed ratio of volatiles

reflects the bulk chemical composition inside the nucleus. The relation be-

twen the two is not simple. It has been realized over the years that the

observed ratio of volatiles in the coma which vary with the heliocentric dis-

tance and also among comets, may be vastly different from those present

in the nucleus even before they were processed by solar radiation. Conclu-

sions on the bulk composition of the nucleus can only be made if the coma

composition is integrated over the entire orbit of the comet.



6.7. Summary

We may briefly summarize the results of the discussion presented so far.

The study of several comets has shown a wide variation of the production

rate of molecules with the heliocentric distance, vastly different from that

of r−2 dependence. In order to explain such a variation, more complicated

models have to be considered. The interesting result is the observed chem-

ical diversity among comets, which has direct bearing on the formation site

of comets. Even though the lines of C2, CN and others dominate the vi-

sual spectral region, their production rates are less by a factor of 100 or so

compared to that of H2O or H. These molecules are, therefore, the minor

constituents of the cometary material. Most of the elements have abun-

dance ratio very similar to the solar atmospheric value except hydrogen

which is very much depleted in comets.

The in situ measurements of Comet Halley made by spacecrafts have

shown the wide variety of ions present in the coma. The production rates of

these species derived for the first time making use of the detailed gasphase

chemistry show them to be a minor contributor of the gas. The presence

of extended source of molecules give strong evidence to show that grains

can also act as suppliers of volatiles in the coma. The expected velocity

distribution of the coma gas based on the detailed modelling agrees with

the Giotto results. Lastly, the spacecraft observations have detected several

possible parent molecules. All the earlier studies indicating H2O as the

parent of OH, H and O have been further reinforced.

Problems

1. Calculate the surface brightenss profiles for a few values of β1/β0 as

shown in Fig. 6.1.

2. Deduce the values of β1 and β0 for the following observed surface bright-

ness distribution: at ρ = 103, 4 × 104 and 7 × 104 km, the observed

values of surface brightness are 350, 35 and 12 kilo Rayleighs respec-

tively. What is the lifetime of the species for an assumed velocity of 1

km/sec?

3. Assume that the diatomic molecule NH is formed due to the colli-

sion process of N and H with α as the rate constant for the reaction.

What is the resulting concentration of NH for nH = 10/cm3, α =

10−10 cm3/ sec and for a time scale of 4× 109 years?



4. Write down the expression for the equilibrium abundances of CH and

CH+ in terms of other abundances for the following reactions: C+H→CH + hν; C+ + H → CH+ + hν; CH + hν → CH+ + e; CH + hν →C + H; CH+ + hν → C+ + H. The rate constants for these reactions

are k1, k2, β1, β2 and β3 respectively.

References

The calculations of surface brightness distribution for the two component

models are discussed in these two papers.

1. Haser, L. 1957. Bull. Acad. Belg.cl. Sc., 5e Ser. 43 740.

2. O’Dell, C. R. and Osterbrock, D. E. 1962. Ap. J. 136 559.

The values required for the calculation of surface brightness distribution

are given in

3. A’Hearn, M. F. and Cowan, J. J. 1975. Ap. J., 80 852.

The three component model is given in the paper

4. O’Dell, C. R., Robinson, R. R., Krishna Swamy, K. S., Spinrad, H. and

McCarthy, P. J. 1988. Ap. J., 334, 476.

For later work refer to

5. Combi, M.R. and Fink, U. 1997, Ap. J., 484, 879.

6. Manfroid, J.M., Hutsemekers, D., Jehin, E. et al. 2007, Icarus, 187,

144.

The vectorial formalism model is presented in

7. Festou, M C. 1981. Astr. Ap., 95, 69.

The Monte Carlo approach and others are discussed in

8. Combi, M. R. and Delsemme, A. H. 1980. Ap. J. 237, 633.

9. Hodges, Jr. R. R. 1990. Icarus, 83, 410.

10. Xie, X. and Mumma, M.J., 1996. Ap.J., 464, 442.

11. Combi, M.R. 1996, Icarus, 123, 207.

12. Crifo, J.-F., et al. 2005. Comets II, eds. M.C. Festou, H.U. Keller and

H.A. Weaver, Univ. Arizona Press, Tucson, p. 471.

13. Combi, M.R. et al. 2005. Comets II, eds. M.C. Festou, H.U. Keller

and H.A. Weaver, Univ. Arizona Press, Tucson, p. 523.

14. Crifo, J.-F. 1990. Icarus, 84, 414.

For some basic parameters, refer to

15. Crovisier, J. 1992. Infrared Astronomy with ISO, eds. Th. Encrenaz

and M.F. Kessler, Nova Science Publishers, p. 221.

16. Huebner, W.F. et al. 1992. Astrophys. Space Sci., 195, 1.



The semi-empirical photometry theory is discussed in the following papers.

17. Newburn, R. L. 1981. In The Comet Halley Dust and Gas Environment.

ESA SP-174, p. 3.

18. Divine, N. et al. 1986. Space Sci. Rev., 43, 1.

The material for prompt emission lines (See. 6.2.2) is from

19. Mamma, M.J. et al. 2001. Ap.J., 546, 1183.

20. Benev, B.P., Mumma, M.J., Dello Russo, N., et al. 2004. Ap.J., 615,

1048.

21. Bonev, B.P., Mumma, M.J., Disanti, M.A., et al. 2006. Ap.J., 653,

774.

Papers referring to water production rates

22. Schleicher, D.G., Woodney, L.M. and Mills, R.L. 2003. Icarus, 162,

415.

23. Schleicher, D.K., Mills, R.L. and Birch, P.V. 1998. Icarus, 132, 397.

Review paper for Sec. 6.2.3.

24. A’Hearn, M.F. et al. 1995. Icarus, 118, 223.

Paper for CN Jet

25. A’Hearn, M.F. et al. 1986. Nature, 324, 649.

Extended source of CO and H2CO

26. Eberhardt, P., Krankowsky, D., Shulte, W. et al. 1987. Astron. Astro-

phys., 187, 481.

27. Meier, R. Eberhardt, P., Krankowsky, D. et al. 1993. Astron. Astro-

phys., 277, 677.

28. DiSanti, M.A., Mumma, M.J., Dello Russo, N. et al. 2001. Icarus,

153, 162.

29. Festou, M. 1999. Space Sci. Rev., 90, 53.

Case of N+2

30. Cochran, A.L. 2002. Ap. J., 576, L165.

31. Cochran, A.C., Cochran, W.D. and Barker, E.S. 2000. Icarus, 146,

583.

The Discussion about complex molecules is drawn from

32. Bockelee - Morvan, D. et al. 2005. Comets II, Eds. M.C. Festou, H.U.

Keller and W.A. Weaver, Univ. Arizona Press, Tucson, p. 391.

33. Crovisier, J. 1987. Astr. Ap. Suppl., 68, 223.

34. Crovisier, J. 2005. Asteroids, Comets and Meteors, eds. D. Lazzaro, S.

Ferraz Mello and J.A. Fernandez, Cambridge University Press, Cam-

bridge, p. 133.

35. Mumma, M.J., Disanti, M.A., Dello Russo, N. et al. 2003. Adv. space

Res., 31, 2563.



36. Disanti, M.A., Villanueva, G.L., Bonev, B.P. et al. 2007. Icarus, 187,

240.

The following paper pertains to Sec. 6.2.9 of oxygen

37. Schultz, D., Li, G.S.H., Scherb, F. and Roesler, F. L. 1992. Icarus, 96,

190; 101, 95, 1993.

For later work

38. Smyth, W.H. et al. 1995. Ap. J., 440, 349.

39. Morgenthaler, J.P. et al. 2001. Ap. J., 563, 451.

40. Oliversen, R.J. et al. 2002. Ap.J., 581, 770.

The following paper refers to discussion of Ly α observations.

41. Keller, H.U. 1976. Space Sci. Rev., 18 641.

Monte-Carlo approach is presented in

42. Combi, M. R. and Smyth, W. H. 1988. Ap. J., 327, 1044.

43. Combi, M. R., Bos, B. J. and Smyth, W. H. 1993. Ap. J., 404, 668.

Some papers pertaining to Hα observations.

44. Smyth, W.H., Marconi, M.L., Scherb, F. and Roesler, F. 1993. Ap. J.,

413, 756.

45. Morgenthaler, J.P. et al. 2002. Earth Moon and Planets, (90), 77.

The basic formulation of gas phase chemistry in the coma can be found in

46. Huebner, W. F. and Giguere, P. T. 1980. Ap. J.,238 753.

For later work

47. Rodgers, S.D. and Charnley, S.B. 2002. MNRAS, 330, 660.

48. Rodgers, S.D. et al. 2005. Comets II, eds. M.C. Festou, H.U. Keller

and W.A. Weaver, Univ. Arizona, Tucson, p. 505.

The following papers give some results based on in situ measurements.

49. Balsiger, H. et al. 1986. Nature, 321, 330.

50. Eberhardt, P. Meier, R., Krankowsky, D. and Hodges, R. 1994. Astr.

Ap., 288, 315.

51. Meier, R., Eberhardt, P., Krankowsky, D. and Hodges, A.R. 1994. Astr.

Ap., 287, 268.

52. Krankowsky, D. 1991. In Comets in the Post-Halley Era, eds. R. L.

Newburn, Jr., et al. Kluwer Academic Publishers p. 855.

53. Keller, H.U. et al. 2005. Comets II, eds. M.C. Festou, H.U. Keller and

W.A. Weaver, University Arizona, Tueson, p. 211.

54. Kissel, J., Krueger, F.R., Silen, J. et al. 2004. Science, 304, 1774.

55. Tuzzolino, A.J., Economou, J.E., Clark, B.C. et al. 2004. Science,

304, 1776.

The following papers may be referred for parent molecules.

56. Krasnopoesky, V. A. 1991. Astr. Ap., 245, 310.



57. Feldman, P.D., Cochran, A. and Combi, M.R., 2005. In Comets II,

eds. M.C. Festou, H.U. Keller and H.A. Weaver, Univ. Arizona Press,

Tucson, p. 433.


CHAPTER 7

Dust Tails

The gaseous material released by the nucleus was studied in the previous

chapter. This material also carries with it a large amount of dust. The idea

that the dust particles are dragged into the tail by the effect of radiation

pressure was first suggested by Bessel in the 1830’s. This concept was

refined by Bredichin around 1900. These ideas were put on a firm basis by

Finson and Probstein in the 1950’s. These authors have worked out in great

detail the dynamics of grains based on fluid and kinetic concepts and the

resulting intensity distributions. Some of these aspects will be considered

in this chapter.

7.1. Dynamics

Consider the case of only pure dust tails. The dust emitted by the

nucleus is subjected to two opposite forces. The solar radiation pressure

acting on the dust particles tries to push it away from the Sun while the

force of solar gravity tries to pull it towards the Sun. Since the two forces

vary as (1/r2), one usually defines an effective gravity, essentially given by

the difference of the two forces. The Keplerian orbit mechanics is then used

for the study of the dynamics of the dust particles. The ratio of radiation

pressure to gravitational force is denoted by (1− µ) where

(1− µ) =Frad

Fgrav. (7.1)

The expressions for Frad and Fgrav for a spherical particle can be written

as

Frad =πd2

4

Qpr

c

(F

4πr2

)(7.2)

213



and

Fgrav =GMr2

(ρdπd

3

6

). (7.3)

Here F is the mean solar radiation field impinging on the grain of diam-

eter d and density ρd. Qpr is the efficiency factor for radiation pressure

(Chap. 8). Therefore Eq. (7.1) becomes

(1− µ) = C(ρdd)−1 (7.4)

where C =3QprF

8πcGM= 1.2× 10−4Qpr.

The nature of the orbit of the particle is decided by the value of (1−µ).

There are two cases of interest that have to be considered. Firstly, particles

are ejected continuously as a function of time by the nucleus. The locus of

the particles which have the same value of (1−µ) is called the Syndyname or

Syndyne curve (Chap. 6). Physically this means that particles of the same

size are emitted by the nucleus. The other case involves the distribution of

particles emitted at any one particular time, as in the case of an outburst.

These particles will have varying (1− µ) values and the locus of the curve

which describes this situation is called Synchrone.

The cometocentric coordinate system used to describe a comet tail is

shown in Fig. 7.1. Here ξ and η represent the coordinates in the orbit plane

of the comet. The third coordinate ζ is in the direction perpendicular to the

plane of the orbit. Let the position of the comet be at A corresponding to

the time of observation t. Consider the position of the comet corresponding

to earlier times say t1, t2, t3 and t4 along its orbit. For simplicity assume

the particles to be emitted from the nucleus with zero relative velocity

and also of the same size. This implies that they have the same value of

(1 − µ). The orbit of the particles can be worked out and the position

on these trajectories for the time of observation t can be calculated. The

final positions are also marked as t1, t2, t3 and t4 in the tail. Therefore the

observed tail is just the locus of the dust particles in the (ξ, η) plane, that

were emitted at different earlier times. This represents the Syndyname or

Syndyne curve. The farther one goes along the tail, the earlier the time at

which the particles were emitted. However, in a real situation, the particles

are emitted with a certain velocity spread, which gives rise to the observed

spread in the tail. The assumption that the particles have the same size

is not very realistic as generally particles of various sizes are emitted at

any instant of time. The actual trajectories of the particles are sensitive to


Dust Tails 215

Fig. 7.1 Trajectories of the particles and the formation of the dust tail.

the particle size as can be seen from Eq. (7.4). The smaller size particles

would have moved farther compared to those of larger size particles. All

these effects have been incorporated in the dynamic model. Some of the

calculated Syndynes and Synchrones are shown in Fig. 7.2.

As mentioned already, in a real situation the dust particles are emitted

continuously as well as with varying sizes from the nucleus for which

(1− µ) ∝ (ρdd)−1. (7.5)

Let Nd(t) represent the dust production rate of all sizes in particles per

second. For simplicity it is assumed that the distribution of particle sizes

remains constant along the orbit of the comet. The particle distribution

function g is given by

g(ρdd)d(ρdd) such that

∫ ∞0

g(ρdd)d(ρdd) = 1. (7.6)



Fig. 7.2 Syndynes and Synchrones for Comet Arend-Roland on April 27, 1957. Timeτ = 1.71 × 106 sec at perihelion on April 8. The position of the observed tail of the

comet is also shown. (This figure as well as the other figures of this chapter are taken

from Finson, M.L. and Probstein, R.F. 1968. Ap. J. 154, 327 and 353).

The quantities Nd(t) and g(ρdd) are usually obtained from the fitting of

the model with the observations.

For the calculation of the orbit of the particle, it is necessary to know the

initial velocities of the dust particles coming out of the nucleus. The model

for the calculation of the velocity of the dust particles is based on the steady

state fluid approach. The flow is assumed to be spherically symmetric with

the same temperature for the gas and dust at the nucleus and dust particles

having a uniform size. The gas and the dust coming out of the nucleus are

coupled through drag forces, computation of which is based on molecular

drag coefficient. With no relative velocity to start with, the particles are

accelerated outwards. The solution of the conservation equations gives the

velocity of the dust particles. These calculations which are quite involved

show that the dust particles reach their terminal speed within about 20

radii of the nucleus. This is indeed a very short distance and hence one

generally uses terminal speed for the dust particles in the tail calculations.

The terminal speed denoted as vi can be expressed in the functional form

asvi

(CpT )1/2= g(M, β) (7.7)

M =md

mgand β =

16

3πρddR0(CpT )1/2/mg


Dust Tails 217

where md is the mass flow rate for the dust and mg that of the gas. There-

fore M represents the dust-to-gas mass ratio. Here R0 is the radius of the

nucleus of the comet, T is the initial temperature of the gas and Cp the

specific heat at constant pressure. Figure 7.3 shows a plot of the velocity of

the dust particles as a function of β. Therefore the velocity can be written

Fig. 7.3 Speed of the dust particles emitted from the inner head region (see text).

in the form

vi = vi(ρdd, Nd, mg) (7.8)

where T and R0 are known. Based on these results, it is generally assumed

that the dust comes out of the nucleus in a spherically symmetric manner

whose speeds are given by the results shown in Fig. 7.3. The results of

more detailed and realistic dust particle size distribution calculations show

that the derived velocities are similar to those in Fig. 7.3, except that the

terminal dust velocity is smaller by about 20% with respect to the single size

dust particles. One can also keep mg, the gas mass flow rate, as a variable.

This value can be obtained by comparing the theoretical calculations with

the observations.

In order to compare with the observed isophotes, the total emission from

the comet along the line of sight has to be calculated. It is necessary to

calculate the integral of the number density along the line of sight, which

is termed as surface density. Since the optical thickness in the dust tail



is small, the surface density is directly proportional to the light intensity.

There are two approaches for calculating the surface density distribution in

the tail. One procedure is to use the particles of various sizes emitted and

then integrate over all the times (Synchrone approach). The other method

is to consider one particle size and then integrate over all the sizes of the

particles (Syndyne approach). The two approaches give the same result as

they involve double integral involving (1 − µ) and time in both the cases.

Only the order of integration in these two cases is reversed. For illustrative

purposes, a brief discussion of the Syndyne approach will be presented.

Consider the position of the comet observed at a particular time tcmeasured from the time of perihelion passage which is taken to be zero.

For a given value of (1− µ), the particles lying on a Syndyne curve would

have been emitted at time

t = tc − τ (7.9)

where τ increases along the Syndyne curve with zero value at the Syndyne

origin. The number of particles emitted in the time τ and τ + dτ and in

the size range (ρdd) and (ρdd) + d(ρdd) is given by

Nd(t)dτg(ρdd)d(ρdd). (7.10)

This has to be multiplied by the light scattering function to get the total

scattered intensity. The scattered intensity is proportional to the cross-

sectional area of the particle and hence varies as (ρdd)2. Working in terms

of (1−µ) rather than (ρdd), the new weighted distribution function can be

written as

f(1− µ)d(1− µ) ∝ g(ρdd)(ρdd)2d(ρdd) (7.11)

such that∫ ∝of(1 − µ)d(1 − µ) = 1. Therefore the Eq. (7.10) can also be

expressed as

Nddτf(1− µ)d(1− µ). (7.12)

These particles then follow well-defined orbits in the plane of the orbit of

the comet. However what is actually seen is the projection of these orbits

in the plane of the sky. Therefore, the expected intensities in the plane of

the sky has to be calculated. The total modified surface density is obtained

by integrating over various Syndyne tails for all values of (1−µ). The final

expression is of the form

D =

∫ (1−µ)b

(1−µ)a

Ndf(1− µ)

[2viτ

dx

dτ(τ ; 1− µ; tc)

]−1

d(1− µ) (7.13)


Dust Tails 219

Here dx/dτ represents the rate of change of length along the given Syndyne

axis with respect to τ . Physically it represents the effect on the surface den-

sity due to dispersion in the longitudinal direction with respect to the tail

axis. On the other hand the other term viτ arises due to the dispersion

in the lateral direction. The Eq. (7.13) involves the functional parame-

ters Nd(t), f(1 − µ) and vi(τ ; 1 − µ; tc). These can be determined from a

comparison of the expected and the observed intensity distributions. From

a knowledge of these three parameters, all other quantities of interest can

then be determined.

The above formalism has been applied very successfully to Comet

Arend-Roland as can be seen from Fig. 7.4. Near perihelion, the gas and

Fig. 7.4 Measured and calculated isophotes for Comet Arend-Roland for May 1, 1957.

D is the density level.

the dust emission rates are about 7.5×107 gm/sec and 6×107 gm/sec. This

gas emission rate corresponds to a value of about 1.5× 1030 molecule/sec.

The dust to gas ratio for Comet Arend-Roland is about 0.8 and the average

size of the particles in the tail is ∼ 1 µm. The particles in the observed dust



tail of Comet Arend-Roland correspond to those which have been emitted

about 10 to 15 days on either side of the perihelion passage. The method

has also been applied successfully to several other comets.

Although Finson and Probstein formalism has been very successful in

the studies of tail morphology, it has several limitations. All the grains are

assumed to be of the same kind and it is difficult to distinguish between

different kinds of grains. The observation of comets has clearly shown a

wide variation in the chemical and physical nature of grains. The value of

Qpr is also not constant but is dependent upon the sizes of the particles. If

the grains of various sizes are emitted continuously, it is difficult to separate

the effects depending on the time of ejection from those due to the size and

properties of the grains. Therefore several studies have been carried out

to take care of some of the above limitations. However, these studies with

complicated numerical integration techniques, loose the great advantage of

the simplicity of the Finson and Probstein approach. But they are highly

successful.

One of these is the Inverse Monte Carlo dust tail model. In this model,

the orbit of sample dust particles are calculated which takes into account the

anisotropy in dust ejection, dust ejection velocity for each dust particle etc.

for modelling the cometary dust tails. The model has been applied to large

number of observed cometary dust tails with good agreement. The model

also calculates the dust mass rate and the time dependent size distribution

function. The time average of size distribution function gives a value of

α = −3.5 for the power law index. The resulting dust size range between

1 µm and about 1 cm.

7.2. Anti-tail

Many comets generally show a short tail in the solar direction which

is called the anti-tail (see Fig. 1.15). The anti-tail really do not point

towards the sun. It is related to the geometry of the system. The theory

developed for the dynamics of the grains of dust tails can also explain the

presence of anti-tail in comets. The calculated Synchrones and Syndynes

for Comet Arend-Roland are shown in Fig. 7.5 which shows the presence of

Synchrones in the direction of the Sun. The size of the particles giving rise

to these synchrons is >∼ 10 µm, which is much larger than those present in

the dust tails which is <∼ 1 µm. The appearance of the anti-tail of Comet

Arend-Roland changed with time within an interval of a few days. This


Dust Tails 221

Fig. 7.5 Calculated Syndynes and Synchrones for tail and anti-tail for Comet Arend-

Roland.

arises due to the changing pattern of the positions of Synchrones. If the

Synchrones are crowded together, then the anti-tail will be seen as a narrow

tail, while if the Synchrones are spread out, the anti-tail will be a broad

one. The visibility of an anti-tail depends mainly on the sun-comet-earth

geometry. It can be seen clearly only when the orbit of the comet cuts the

orbital plane of the Earth.

The anti-tail of Comet Kohoutek could be seen shortly after perihe-

lion. Comet Halley also displayed anti-tail after its perihelion passage. In

general, the anti-tail is more favourable for observation after perihelion.

The dynamical theory of grains has been applied very successfully to the

observed anti-tail of many comets. In fact, the predictions of the anti-tail

based on the theory have later been confirmed through observations. The

large size grains present in the anti-tail is also consistent with the estimated

size of the particles based on the absence of 10 µm emission feature in the

anti-tail of Comet Kohoutek (Chap. 9). The presence of anti-tail of comets

is therefore generally explained as due to the effect of large size particles.



It is evident from the dynamical theory of grains that the size of the

particles along the tail should vary and it depends upon the nature of

Synchrone and Syndyne curves. Since the polarization measurements are

sensitive to particle sizes, they could effectively be used for determining the

sizes of the particles. The polarization measurements on comets do seem

to indicate the variation of the particle size along the tail of the comet.

7.3. Dust Trails

Dust trails were first seen on Comet Tempel 2 from the observations

carried out with the Infrared Astronomical Satellite. They are very narrow

with length to width ratio of around 200 to 1. Many of the trails are found

to coincide with the orbit of comets. Dynamical calculations show that

the observed dust trails are coincident with large size particle trajectories

indicating the presence of particles in the size range of about millimetre to

centimetre. These particles are ejected out at low velocities ∼ a few m/s

from the nucleus. The narrowness of the trail indicate that the velocity

dispersion is quite low and are therefore stable against perturbing forces

over periods of atleast a few orbital revolutions. Therefore the observed

dust trails is the resultant effect of the build up of large particle emissions

over several orbits. The observations carried out with ISO could spatially

resolve the dust trail in Comet Encke. Because of very large particles

present in dust trails, it may be too faint to be seen in the visible region.

However it has been detected along the orbit of Comet Kopff in the visible

region when the comet was at r = 3.01 AU. Therefore study of dust trails

in comets in the visible region should help in a better understanding of the

nature of very large size particles.

7.4. Sodium Gas Tails

A narrow, straight and long tail along with a diffuse tail superposed over

the dust tail composed of sodium atoms has been seen in the bright Comets

Hale-Bopp and Hyakutake. These were seen in the images taken with filters

centred on the sodium D lines at 5890 A. These are similar to plasma tail

and dust tail that are generally seen in comets. The straight sodium tail

extending upto ∼ 107 km is located between the ion tail and the prolonged

radius vector. The straight tail could arise from the dissociation of some

molecule containing sodium which is then dragged out due to radiation


Dust Tails 223

pressure. The diffuse sodium tail could arise by the release of sodium

from dust grains due to some physical processes, such as evaporation and

sputtering. It is possible that sodium tails could be a common feature of

comets.

7.5. Dust features

The high quality photographs near the nuclear region show highly com-

plicated structure of various kinds as can be seen from Fig. 7.6. This arises

Fig. 7.6 The top rows show the various types of features observed in six comets in thevicinity of the nucleus. The bottom rows show the computer simulated images. There isa good similarity between the two. The Sun is at the top. (Sekanina, Z. 1991. In Comets

in the Post-Halley Era, eds. R.L. Newburn, Jr., et al. Kluwer Academic Publishers, p.769).



due to the complicated flow pattern of the grains after they are released

from the nucleus. To a first approximation it can be described in terms

of the resulting force between the gravitational, Fgrav and radiation Frad,

forces acting on the grains (Sec. 7.1). The trajectories of the grains under

such a force in a cometocentric system are approximately parabolic in na-

ture. Different patterns could be produced depending on the parameters

of the grains like radius, density and the material property (Fig. 7.7). In a

Fig. 7.7 Dust particle trajectories for Comet Halley released from the nucleus at the

time of perihelion passage. (a) and (b) correspond to particle sizes of 10µm and 0.8 mmrespectively. The dotted curve shows the resulting envelope (Fertig, J. and Schwehm, G.

1984. Adv. Space Res. 4, 213).

real situation the spin of the nucleus, its associated precession and nutation

and the cometary activity has to be considered. These will give rise to com-

plex behaviour of the flow patterns of the dust particles in the near nucleus

region. For example, arising from a single active area on the nucleus, the

spiral jets unwind and develop into expanding parabolic envelopes and so

on. It is rather difficult to take into account all these effects in the model

calculations. Therefore several attempts have been made to understand the

observed features in terms of simple hydrodynamic models. Another ap-

proach that has been attempted is to understand the flow patterns through

computer generated synthetic images by varying the parameters. As can

be seen from Fig. 7.6, the computer simulation technique can in principle

reproduce the diverse patterns of the observed dust structures of comets.

However the disadvantage of this method is that there are many parameters

involved which have to be adjusted by trial and error method to match the

observations, such as the angles associated with spin axis positions and its

relation to Earth and Sun, rotation period, position of active regions, dust


Dust Tails 225

properties and so on. Therefore it is possible to have multiple solutions in

this approach. To limit the number of solutions, it is preferable to fix some

of the parameters based on some other observations.

Several other fine features have also been seen in comets. The streamers

seen are straight and curved structures which converge towards the nucleus.

They are generally associated with synchrones. For example the streamers

seen in Comet West (1976 VI) have been attributed to particles of different

sizes expelled simultaneously from the nucleus in the form of an outburst.

Therefore long streamers are indicative of large dispersion of particle sizes,

whereas a narrow width reflects a short duration outburst. Several well

separated dust streamers has been seen in the images of Comet Halley. The

analysis of these streamers is consistent with the interpretation that they

represent diffuse synchrones of dust particles. The time interval between

these streamers is found to be around 2.2 days which corresponds to the

rotational period of the nucleus.

Another kind of feature, called striae has been seen in several comets.

These are characterized by a series of parallel narrow bands at large dis-

tances from the nucleus. They do not start from the nucleus. The mecha-

nism of formation of striae is not clear at the present time.

The dynamical theory developed to explain the shape of the dust tail

of comets has been very successful in explaining the observations. In par-

ticular it can explain quantitatively the observed isophotes of dust tails of

comets. It has also been possible to get production rates of the gas and dust

and therefore the dust to gas ratio. The theory is also very successful in

explaining the presence of anti-tail in comets. Several sophisticated models

have been carried out over the simple dynamical theory, which can explain

in detail the observed dust tail of comets.

Problems

1. A grain of size a and density ρ is moving in a medium of hydrogen den-

sity NH due to the effect of radiation pressure of a star of temperature

T and radius R. The grain is being slowed down by the drag force due

to the hitting of the grain and sticking to it. Show that the terminal

speed of the grain is given by

vt =

(R2σT 4

mHc

)1/21

N1/2H r

.



Hint: Use the conservation of momentum of the particle for calculating

the drag force.

2. Why does the length of the dust tail change with heliocentric distance?

At what distance from the Sun will the tail have the maximum extent.

3. How does the tail of a comet look, when it passes by Jupiter at a close

distance?

References

The theory of dynamics of dust tails is worked out in detail in these papers

1. Finson, M.L. and Probstein, R.F. 1968. Ap. J. 154, p. 327. (Paper

I), p. 353 (Paper II).

Other relevant papers are the following

2. Crifo, J.F. 1991. In Comets in Post-Halley Era, eds. R.L. Newburn,

Jr. et al., Kluwer Academic Publishers, Vol. 2, p. 937.

3. Fulle, M. 1989. Astr. Ap. 217, 283.

4. Fulle, M. 2005. In Comets II, eds. M.C. Festou, H.U. Keller and W.A.

Weaver, Univ. Arizona Press, Tucson, p. 565.

5. Grun, E. and Jessberger, E.K. 1990. In physics and chemistry of

comets. ed. W.F. Huebner. Springer-Verlag. p. 113.

6. Richer, K. and Keller, H.U. 1987. Astr. Ap. 171, 317.

7. Sekanina, Z. and Farrell. J.A. 1978. A.J 87, 1836.

Dust Trail

8. Eaton, N., Davis, J.K. and Green, S.F. 1984. MNRAS, 211, 15.

9. Sykes, M.V., Lebofsky, L.A., Hunten, D.M. et al. Science, 1986. 232,

1115.

Detection in the visible for the first time

10. Ishiguro, M., Watanabe, J., Usui, F. et al. 2002. Ap. J., 572, L117.

Cometary dust features can be seen in

11. Rahe, J., Donn, B. and Wurm, K. 1969. Atlas of Cometary Forms,

NASA SP-198 U.S. Govt. Printing Office, Washington D.C.

The following papers discuss model calculation of features

12. Vasundhara, R., Chakraborty, P., Muneer, S. et al. 2007. A.J., 133,

612.

13. Sekanina, Z., Brownlee, D.E., Economou, T.E. et al. 2004. Science,

304, 1769.


CHAPTER 8

Light Scattering Theory

The dust particles present in comets scatter and absorb the incident

solar radiation. In fact, it is the scattered solar radiation which makes the

dust tail of comets visible. The infrared radiation that has been seen from

comets is also directly related to the amount of incident radiation absorbed

by the dust particles. Therefore the theory of scattering of light by small

particles is basic to the study of cometary grains. The aim is to determine,

from theory, the distribution of intensity of the scattered radiation and the

polarization as a function of the scattering angle. It is also of interest to

know the cross sections for the absorption and scattering processes, which

determine the albedo of the particles. The efficiency factor for the radia-

tion pressure is also of interest, as was discussed in the last chapter. All

these quantities depend upon the shape, structure and composition of the

grain. The theories of scattering have been developed for well-defined par-

ticle shapes like spheres, concentric spheres, cylinders, spheroids and so on.

However, grains in general are likely to be irregular in shape, inhomoge-

neous and fluffy in nature. The theory of scattering from such grains is

highly complex. Hence, at the present time attempts are being made to

study the scattering from such grains from the theoretical points of view

with certain approximations. The study is also being pursued through ex-

perimental means. We will briefly discuss some of these aspects in this

chapter.

227



8.1. Mie Scattering Theory

8.1.1. Efficiency factors

The theory of scattering by spherical particles of homogeneous composi-

tion involves the solution of Maxwell’s equations with appropriate boundary

conditions on the sphere. This was worked out by Mie in 1908 and indepen-

dently by Debye in 1909. Since an excellent account of the solution of this

scattering problem, generally known as Mie Theory, is discussed in many

books, we will summarize here only the results with particular reference to

the case of cometary dust.

The scattering properties of a particle depend upon the following quan-

tities: (1) the property of the medium, usually specified by the complex

refractive index, m = n− ik. where n and k are the refractive and absorp-

tive indices respectively. (2) the wavelength of the incident radiation (λ)

and (3) the size of the particle (a). As a result of radiation interacting with

the particle, part of the radiation is absorbed and part of it is scattered.

Therefore, the total amount of radiation lost from the incident beam (ex-

tinction) is the sum total of the absorbed and the scattered components.

These are generally expressed in terms of the dimensionless efficiency fac-

tors, Qsca and Qabs for the scattering and absorption components. The

efficiency factor for the total extinction is given by

Qext = Qsca +Qabs. (8.1)

If Csca, Cabs and Cext denote the corresponding cross sections, then

Csca = πa2Qsca (8.2)

Cabs = πa2Qabs (8.3)

and

Cext = πa2Qext. (8.4)

The main aim is to determine the efficiency factors Qsca and Qabs from

the scattering theory. They are given by

Qsca =2

x2

∞∑n=1

(2n+ 1)|an|2 + |bn|2

(8.5)

Qext =2

x2

∞∑n=1

(2n+ 1)Re(an + bn) (8.6)


Light Scattering Theory 229

where the dimensionless parameter x = (2πa/λ) and Re represents the real

part. The scattering coefficients an and bn are given by

an =ψ′n(mx)ψn(x)−mψn(mx)ψ′n(x)

ψ′n(mx)ζn(x)−mψn(mx)ζ ′n(x)(8.7)

bn =mψ′n(mx)ψn(x)− ψn(mx)ψ′n(x)

mψ′n(mx)ζn(x)− ψn(mx)ζ ′n(x). (8.8)

ψn and ζn are the modified Bessel functions known as the Riccati-Bessel

functions. The addition of a prime to the Riccati-Bessel functions denotes

the differentiation with respect to their arguments. Riccati-Bessel functions

can be expressed in terms of Bessel functions, J , as follows

ψn(y) =(πy

2

)1/2

Jn+1/2(y) (8.9)

and

ζn(y) =(πy

2

)1/2 [Jn+1/2(y) + i(−1)nJ−n−1/2(y)

](8.10)

where y represents either mx or x. The third Riccati-Bessel function can

be defined as

χn(y) = (−1)n(πy

2

)1/2

J−n−1/2(y) (8.11)

The functions ψn(y) and χn(y) are connected through the identity

ζn(y) = ψn(y) + iχn(y). (8.12)

In addition to energy, light carries momentum. Therefore a beam that

interacts with the particle will exert a force on the particle called radiation

pressure. With the assumption that all the photons absorbed by the particle

transfer all their momentum and hence exert a force in the direction of

propagation, the efficiency for radiation pressure is given by the difference

between the total extinction and the amount of scattered light.

i.e. Qpr = Qext − cos θQsca (8.13)

where

cos θ Qsca =4

x2

x∑n=1

n(n+ 2)

n+ 1[Re(an)Re(an+1)

+ Im(an)Im(an+1) + Re(bn)Re(bn+1)

+Im(bn)Im(bn+1)] +2n+ 1

n(n+ 1)[Re(an)Re(bn) + Im(an)Im(bn)]

(8.14)



where Re and Im represent the real and imaginary quantities. The ef-

ficiency for radiation pressure is of interest in the study of dynamics of

grains. The value of cosθ is also sometimes known as the asymmetry fac-

tor.

i.e. g = cos θ. (8.14a)

The value of g = 0 for a particle that scatters isotropically and g > 0

for preferentially scattering in the forward direction. For complete forward

scattering g = 1. For g < 0 the scattered radiation is in the backward

direction. The cross sections given by Eqs. (8.5) and (8.6) refer to a particle

of a given size. However, in general, a medium will consist of particles of

various sizes. If n(a) represents the number of particles per unit volume

in the size range a and a + da then the total extinction is given by the

expression

Qtotal(λ) =

∫πa2Qext(a, λ)n(a)da. (8.15)

8.1.2. Albedo

Another quantity of interest is the amount of energy scattered from

the incident beam, called the albedo of the particle. In the general case,

when the scattered contributions from diffracted, refracted and reflected

components are taken into account, the albedo of the particle is defined as

γ =Qsca

Qext. (8.16)

If the diffraction component is not taken into account, it is called the bond

albedo. Therefore the bond albedo is defined as the ratio of the energy

refracted and reflected by the particle in all directions to the energy incident

on the geometric cross section.

i.e. AB =1

G

∫σr(θ, φ)dω (8.16a)

Here σr(θ, φ) is the differential scattering cross-section. For the case

when there is symmetry with respect to the direction of incident radiation,

then

σr(θ, φ) = σr(θ).

The geometric albedo Ap is defined as the energy scattered by the par-

ticle at θ = 180 (backward scattering) to that scattered by a white disk



of the same geometric cross section, scattering according to Lambert’s Law

(σr(180) = G/π).

i.e. Ap =π

Gσr(180). (8.16b)

The bond albedo AB is related to the geometrical albedo Ap by the relation

AB = Ap.q (8.16c)

where

q = 2

∫ π

0

j(θ) sin θdθ.

Here j(θ) = σr(θ)/σr(180) ≡ F (θ)/F (180), the ratio of the phase func-

tions and q is known as the phase integral. For isotropic scattering, AB = 1

and Ap = 0.25 and therefore q = 4.

8.1.3. Scattered intensity

The major physical quantity of interest is the intensity of the scattered

radiation as a function of the scattering angle. If I0 is the original inten-

sity impinging on the grain, the intensity of the light scattered into unit

solid angle for the scattering angle θ defined with respect to the incident

beam (Fig. 8.1) is given by F (θ)Io, where F (θ) denotes the phase func-

tion. Therefore a plot of F (θ) versus θ gives the angular distribution of the

Fig. 8.1 Schematic diagram of a scattering process. θ is the scattering angle andIo(θ)F (θ) is the intensity of the scattered radiation from the original incident radia-tion of Io.

scattered radiation and is known as scattering diagram. The scattering

phase function is related to the complex scattering amplitudes s1(θ) and

s2(θ) as

F (θ) =1

2k2

[|s1(θ)|2 + |s2(θ)|2

]= i1 + i2 (8.17)



where k = (2π/λ). The quantity i1 and i2 are essentially the components of

intensity in the direction perpendicular and parallel to the scattering plane.

The scattering plane contains the incident radiation and the direction of

the scattered wave. The expressions for s1(θ) and s2(θ) are given in terms

of the scattering coefficients an and bn as

s1(θ) =∞∑n=1

2n+ 1

n(n+ 1)anπn(cos θ) + bnτn(cos θ) (8.18)

s2(θ) =∞∑n=1

2n+ 1

n(n+ 1)bnπn(cos θ) + anτn(cos θ) (8.19)

where

πn(cos θ) =1

sin θP 1n(cos θ) (8.20)

and

τn(cos θ) =d

dθP 1n(cos θ). (8.21)

Here Pn’s are the Legendre polynomials.

8.1.4. Polarization

The scattered intensities i1 and i2 in the two directions are a function of

the scattering angle. Therefore the polarization of the resulting radiation

is defined as

P =i1 − i2i1 + i2

. (8.22)

The value of |P | varies from 0 to 1. The sign of P could be positive or

negative. Positive and negative signs imply that the scattered light is po-

larized perpendicular or parallel to the scattering plane respectively. The

value of P for the scattering angles, θ = 0 and 180, is equal to zero as

i1 = i2. In addition to linear polarization, circular polarization also may

be seen in certain favourable cases. The circular polarization arises if the

refractive indices are different for the two states of polarization. In general,

the circular polarization is one or two orders of magnitude smaller than

that of linear polarization.



8.2. Approximate Expressions

So far we have been considering the exact expressions for the efficiency

factors for the scattering and the extinction by the spherical particles. In

many practical situations, like in the infrared and far infrared regions, where

the condition that the size of the particle is very much less than the wave-

length of the radiation, i.e., x <∼ 1 is satisfied, the approximate analytical

expressions of Eqs. (8.5) and (8.6) may be used. Under the above condition,

the Eqs. (8.5) and (8.6) reduce to

Qsca =8

3x4Re

m2 − 1

m2 + 2

2

(8.23)

and

Qabs = −4xIm

m2 − 1

m2 + 2

. (8.24)

If further m is real, i.e. no absorption, then

Qsca = Qext ≈8

3x4|m

2 − 1

m2 + 2|2 (8.25)

and Qabs = 0.

8.3. Computation of Cross Sections

The actual evaluation of the efficiency factors has been considered in de-

tail by various investigators. Various methods have been proposed for the

calculation of efficiency factors starting from the simple expansion tech-

niques. The main problem is to calculate the coefficients an and bn and

this is quite tedious. The calculation of an and bn involves a knowledge

of the quantities ψn, ψ′n and others which in turn depend upon the Bessel

functions with complex arguments. The computation of these quantities be-

comes complicated when the refractive index is complex. However they can

be calculated easily using recurrence relations with the help of a computer.

The following recurrence relations can be used to derive the Riccati-Bessel

functions.

ψn(y) =2n− 1

yψn−1(y)− ψn−2(y) (8.26)

ψ′n(y) = −ny

ψn(y) + ψn−1(y) (8.27)



χn(y) =2n− 1

yχn−1(y)− χn−2(y) (8.28)

and

χ′n(y) =n+ 1

yχn(y)− χn+1(y) (8.29)

with

ψ0(y) = sin y (8.30)

ψ1(y) =sin y

y− cos y (8.31)

χ0(y) = cos y (8.32)

and

χ1(y) =cos y

y+ sin y. (8.33)

Therefore all the functions that enter in an and bn can be generated.

Hence the efficiency factors can be computed for any given value of m and

x. The number of terms to be included in Eqs. (8.5) and (8.6) depend upon

the parameters x and m. If x is large, more terms have to be included in the

calculation of an and bn. However with the availability of computers it is

easy to sum over all the terms till the required accuracy is achieved in the

calculated efficiency factors. Many investigators have developed efficient

computer programs for the calculation of these quantities which are readily

available in the literature.

8.4. Results

For astrophysical situations, one seems to deal with the real part of

the refractive index of the particles, which is of the order of 1.3 to 1.6

(Chap. 9). Therefore, for illustrative purposes, the extinction cross-section

is plotted as a function of x for constant indices of refraction of 1.3 and

1.6 in Fig. 8.2. Since the indices of refraction are assumed to be constant,

the nature of the curves remains the same when the size or wavelength is

changed. But in a real situation, the refractive index of the material is a

function of the wavelength which has to be incorporated. The results of such

calculations give a better physical picture and show more structures in the

curves arising due to the material property. The curves in Fig. 8.2 display

several interesting features. A series of broad maxima and minima can be



Fig. 8.2 The efficiency factors for extinction and scattering for spheres calculated from

the Mie Theory are plotted as a function of the dimensionless parameter X. The results

are shown for three sample cases. (Adapted from Wickramasinghe, N.C. 1973. Lightscattering functions for small particles. London: Adam Hilger).

seen easily. Superposed on these band features are the small irregular fine

structures generally called the ripple structure. The curve for small values

of x refers to Rayleigh scattering region, where it varies as (1/λ4). For

intermediate values of x, the variation is given by (1/λ). For large values of

x, the value of Qext approaches a value of 2. It is interesting to note that the

limiting value 2 is twice as large as the cross section given by the geometrical

optics. This is essentially related to the fact that the geometrical optics is

strictly not valid in the neighbourhood of the edges of the object. So one

has to use the diffraction theory. The additional factor of πa2 comes due to

this reason. The effect of absorption in the Qext is to reduce the strength

of the resonances and the curve becomes smooth (Fig. 8.2). For very large

values of x, the absorption and scattering contribute almost equally while

for small values of x, the total extinction is mainly contributed by the

absorption process. The effect of increase in the absorption coefficient is to

reduce the albedo of the particle.

The scattering intensity for various angles is shown in Fig. 8.3. The



Fig. 8.3 The intensities I1 (dashed curve) and I2 (solid curve) are plotted as a functionof the scattering angle for two representative values of the refractive index. Each curve

is labeled by the parameter X. (Adapted from Wickramasinghe. N.C. 1973. op. cit.).

values of the intensity at 0 and 180 are marked in the margin of the curve.

The most important point to note from the curve is that the scattering is

mostly in the forward direction (θ = 0) and it oscillates with the scattering

angle. The scattering diagrams for a few cases are shown schematically in

Fig. 8.4. For isotropic scattering the intensity is the same in all directions.

For Rayleigh scattering i.e., particle size <∼ wavelength of the incident

radiation, the phase function ∝ (1 + cos2 θ) where θ is the scattering angle.

This dependence shows that the resultant diagram shown by the solid line

is a superposition of two different components. One component which is

isotropic, is shown by dashed lines and the second component is shown by

two lobes. The Mie scattering diagram for m = 1.3, x ≈ 1 and x > 1 is

also shown in Fig. 8.4. Here as x increases, the scattering becomes more



and more forward peaked.

Fig. 8.4 Schematic scattering diagrams for various cases.

However, in any real situation, there will be a distribution of particle

sizes. Therefore the cross-sections have to be averaged over the size distri-

bution function. Several analytical expressions have been used for averaging

the cross-sections. For the purpose of illustration Fig. 8.5 shows the results

for the size distribution of the form

n(r) = constant r(1−3b)/be−r/ab (8.34)

where a = reff and b = veff . Here reff and veff represent a mean radius

for scattering and a measure of the width of the size distribution function

respectively, b = 0 refers to the case of single particles. The effect of the size

distribution function is to average out the ripple structure which smoothens

it out. It also dampens the resonance effects. Therefore the curve becomes

smooth. The results for phase function and linear polarization are shown

in Fig. 8.6. Although the results refer to a particular size distribution

function, the general trend and nature of the results are similar for any

other size distribution function.



Fig. 8.5 A plot of the efficiency factor for scattering as a function of the dimensionless

parameter X = (2πa/λ). The curves represent the averaged value summed over thesize distribution of the form n(r) = const. r(1−3b)/be−r/ab. The refractive index is

n = 1.33, k ≡ ni = 0. (Hansen, J. E. and Travis, L.D. 1974. Space Sci. Rev. 16 527).

8.5. Particles of Other Types

So far, the discussion has been limited to the case of spherical particles

with homogeneous and isotropic internal structure. Even in this case the

mathematical solution is highly complex in nature. However in a real situa-

tion, the particles are far from being spherical in shape. Therefore consider-

able effort has been spent in the study of particles of other shapes through

theoretical and numerical means. Several classical methods of solving scat-

tering problems have been used. To name a few, separation of variables,

perturbation methods, fields expanded in vector spherical harmonics among

others have been used. The exact solutions for scattering from spheroids,



Fig. 8.6 Phase function and polarization for single scattering are plotted as a function

of the scattering angle for wavelength of 5500 A and n = 1.33, k ≡ ni = 0 (Adapted

from Hansen, J. E. and Travis, L. D. 1974. op. cit.).

infinite cylinders, concentric spheres have been worked out and are avail-

able in many books. The cases of cylinders with oblique incident, as well

as concentric cylinders have also been investigated mathematically. The

approximate formulae for the limiting cases are also available for concen-

tric spheres, cylinders etc. However, even the well defined shape for the

particles introduces many more additional parameters due to their shape,

orientation etc. compared to that of spheres.

Two possibilities exist for the case of spheroids. They could be prolate

or oblate. For infinite cylinders, the exact solution has been derived for

the case when the particle diameter is much smaller than its length. In the

case of concentric spheres there are various possibilities depending on the

inner and outer radii and their relative refractive indices. Even from such

well-defined shape of particles the computation of cross sections is quite

complicated and lengthy. In addition, the computation time could be quite

large and particularly so for larger size particles. Therefore efforts are being

made to develop efficient methods for the calculation of cross sections from

such well-defined particles.

The general results for nonspherical cases are qualitatively very similar

to those for spheres, with a Rayleigh like increase, broad scale interference

structure with finer ripples superposed over it. As a typical case, the results

for the extinction efficiencies in the case of infinite cylinders for n = 1.33

and m = 1.33 − 0.05i are shown in Fig. 8.7. Here QE and QH denote



Fig. 8.7 A plot of the efficiency factors for extinction of cylinders oriented normal to

the incident radiation for two values of refractive index. QE is for the electric vectorE of the incident radiation parallel to the axis and QH is for the magnetic vector H of

the incident radiation parallel to the axis. The polarization is also shown as QE −QH .(Adapted from Greenberg, J. M. 1978. In Cosmic Dust. ed. McDonnel, J. A. M. New

York: John Wiley and Sons, p. 187).

the value of Q for the case of electric vector and the magnetic vector of

the incident radiation parallel to the axis of the cylinder. The radius of

the cylinder is ‘a’. The polarization (QE −QH) is also shown in the same

figure. The effect of absorption as pointed out before, is to dampen the

polarization oscillations.

The discussion so far was limited to the ideal situations wherein the

particles are assumed to have definite and regular shapes. However, in gen-

eral, the dust particles in a real case are far from the above situations. It is

more likely that they will be irregular in shape. They could also be fluffy,

porous and could have surface roughness. Therefore several attempts have

been made to consider the general case of scattering by an arbitrary parti-



cle. The general solutions involve the calculation of the effect on individual

atoms by the incident field and also the combined fields of all other atoms.

As the general case is almost impossible to solve, it necessarily involves

making certain approximations and finally the cross sections have to be de-

rived from numerical methods. Several studies have been carried out based

on perturbation approach, T-matrix method also known as the extended

boundary condition method, study of statistical description of roughness in

the frame work of the potential theory and so on.

However a method which has been successfully used is to subdivide the

particle into several smaller identical elements, which itself is a collection of

large number of atoms, and can be represented as a dipole oscillator. The

average field due to the combined effect of all these individual oscillations

is then calculated. This is the well known Discrete Dipole Approximation

(DDA) method. Another approach that has been investigated is to make

use of the integral representation of the macroscopic Maxwells equations.

In this method, the particles are assumed to be an aggregate of small cubic

volumes at the lattice site of which are located sub volumes, of identical

and homogeneous character. This is mathematically equivalent to that of

DDA except that the computation is based on the calculation of volume

element polarizations. The advantage of DDA method is that it can be

applied to particles of arbitrary shape, structure and composition. A sample

representation of a fluffy particle in terms of subparticles which is considered

in the computation of cross sections is shown in Fig. 8.8. The nature of

fluffiness i.e. creating porosity in the grain, depends upon the number of

dipoles, their sizes and the separation between them. The number of dipoles

used in the calculation could be as large as 105. Each of the dipoles in the

grain could be small enough to be thought of as Rayleigh-limit inclusions.

Extensive calculations have been carried out for the study of scattering

properties of such a grain under the DDA formalism for examining the

effects of porosity, size of the dipoles, topology of the grain and so on,

on the derived results. A sample result for the extinction from a prolate

spheroid is shown in Fig. 8.9. The validity of DDA results has to be tested

by comparing the results with those derived from well-defined shapes.

In the past, the scattering from regular particles like spheres, spheroids

etc., but with inhomogeneous composition, has been studied by replacing

the inhomogeneous grain with a homogeneous grain by using an effective

dielectric constant. The idea being that the effective dielectric constant of

the material provides the correct light scattering characteristics of the het-

erogeneous medium. It can be calculated either from the Maxwell-Garnett



Fig. 8.8 The model of the dust particle (a prolate pseudospheriod with 2a = 60, 2b =

2c = 30 in units of the lattice spacing d) used in the calculation of cross-section underDiscrete-Dipole approximation. In this grain model 60% of the original 28256 dipoles

have been removed (Wolff, M. J., Clayton, G. C., Martin, A. G. and Schulte-Ladbeck,

R. E. 1994. Ap. J. Letters, 423, 51).

theory or the Bruggeman theory (Sec. 8.6). It is therefore of interest to com-

pare the results obtained from such calculations with those derived from

DDA formalism, for the same grain parameters. Figure 8.9 shows such a

comparison. The agreement in general is found to be better, as the number

of dipoles in the DDA formalism is increased. Ultimately, how good are

the calculations has to be tested by comparing with the observations. In

general, the results based on these theories seem to represent astronomical

observations better compared to the case of spherical particles. However

the results based on spherical dust particles can be used to limit the range

of cometary grain parameters.

It is easier to carry out laboratory measurements on larger size particles

rather than on micron size particles. Therefore several studies have been

carried out using the microwave techniques. If the optical constants of the

material in the microwave and optical wavelength regions are the same, then



Fig. 8.9 The calculated efficiency factors for extinction with 40% porosity. The resultsare for two orientations of the incident electric vector E and H which are parallel and

perpendicular to the major axis of a 2:1 prolate spheroid. The DDA results for both

Rayleigh and non-Rayleigh vacuum inclusions are compared to the various mixing rules(continuous curves). The base grains have 3558, 8320 and 14440 dipoles for the ranges

x ≤ 2, 3 ≤ x ≤ 4 and x ≤ 5 respectively (Wolff, M. J. et al. 1994. loc. cit.).

the results based on the microwave studies carried on larger size particles

are also valid for smaller size particles in the visible wavelength region,

as the characteristic parameter x is the same. Therefore, various kinds

of measurements like extinction, angular scattering and so on have been

carried out in the microwave region on particles of various shapes such as

cylinders, spheroids, spheres and irregular particles. The results of such

studies agree reasonably well with those of calculated values, thus giving

some credibility for the theories.

8.6. Optical Constants

The calculation of cross sections requires the knowledge of the gross

material property of the grain. This is generally represented by the re-

fractive index, m = n + ik, or the dielectric constant ε = ε′ + iε′′, of the

medium. The optical constants represent a measure of the capacity of elec-

trons in the material to oscillate for the incident radiation. m and ε are

generally complex quantities. These two quantities are not independent of

each other. Knowing one set of quantities, the other set can be calculated.

The relations for ε′ and ε′′ are given by

ε′ = n2 − k2 and ε′′ = 2nk. (8.35)



similarly the relations for n and k are given by

n =

[(ε′2 + ε′′2)1/2 + ε′

2

]1/2

(8.36)

and

k =

[(ε′2 + ε′′2)1/2 − ε′

2

]1/2

. (8.37)

The determination or the use of either refractive index or the dielectric

constant depends upon the physical situation being considered.

The optical constants of materials are not directly measurable quan-

tities. But they have to be determined from laboratory measurements of

some property of light in combination with the suitable theory. Various

types of measurements can be made for the determination of the optical

constants of the material. They could be the measurement of the extinc-

tion coefficient from a set of thin slabs, transmittance and reflected light at

near normal incidence or for various incidence angles, measurement of the

phase shift of the reflected light and so on. The theoretical formulation for

the interpretation of the above types of measurements are available. The

choice of the method depends upon the material to be measured, experi-

mental techniques available and so on. Various methods have been used to

derive the optical constants of several types of material. However the most

commonly used and probably the best method for the determination of

optical constants is the use of dispersion relations, generally known as the

Kramers-Kroning relations in combination with the reflectance measure-

ments. Basically, the dispersion relation is an integral over the whole range

of photon energies, which relate the dielectric function to the measured

quantity like the extinction coefficients or the reflectance measurements.

The real and imaginary parts of m or ε are not independent of each other

but are interrelated through these dispersion relations.

The determination of optical constants of materials from the laboratory

measurements are quite difficult and cumbersome. The sample has to be

perfectly smooth to avoid scattering effects, it should be homogeneous and

thin slab and so on. Most of the optical constants derived from laboratory

measurements are of that of pure substances. In addition the wavelength

coverage may not be extensive. Therefore one might have to make use of

the derived optical constants over the limited wavelength regions available,

to cover the wavelength region from ultraviolet to far infrared. The optical

constants for several material like graphite, carbon, amorphous carbon,

silicates and so on which are of interest for cometary studies are available.



The optical constants for several terrestrial rock samples and moon samples

have also been measured. The measured refractive indices of silicates for

different Mg/Fe ratio show wide variation. This will have great effect on

grain temperature and hence on the infrared emission from such grains.

Since these materials are not strictly homogeneous, the measurements in

principle, should provide some sort of average dielectric constants of these

materials. The cometary grains are inhomogeneous in character. Therefore

the best way to get the refractive index of the material of interest is through

laboratory studies. Since it is almost impossible to study all the cases of

interest in the laboratory, a theoretical treatment of the problem becomes

a necessity.

The dust grains present in the astrophysical environments are highly in-

homogeneous in character. In general, it is not easy to calculate the average

optical properties of inhomogeneous medium even if the property of the in-

dividual constituents are known. So one has to use various approximations.

This has given rise to the concept of an effective dielectric constant for the

material. This has given rise to several theories. The two theories that are

most commonly used are the Maxwell-Garnett theory dating back to 1904

and the Bruggeman theory proposed in 1935. In the Maxwell-Garnett the-

ory, the composite grain consists of matrix material in which small spherical

inclusions are embedded having dielectric constants εm and εi respectively

with f as the volume fraction of inclusions. The inclusions are assumed to

be spherical in shape with sizes smaller than or nearly equal to the wave-

length under consideration. The problem is to calculate the average electric

field of the medium from the knowledge of the individual electric fields Eiand Em of the two components arising out of the interactions, which are by

themselves are averages over a smaller volume. With the assumption that

the relations for the electric fields, polarization vector and the relations

between Ei and Em as well as between εi and εm also hold for the average

fields, the following relation for the average dielectric constant is derived.

εav = εm [(2εm + εi − 2f(εm − εi))/(2εm + εi + f(εm − εi))] . (8.38)

Here the average dielectric constant is independent of the position in

the composite medium. This is generally known as the Maxwell-Garnett

rule or simply as the M-G rule.

In the Bruggeman theory, there is no distinction between the matrix and

the inclusions as in the case of Maxwell-Garnett theory. Here the grains

are aggregate of small particles with their own dielectric constant (εi) and



volume fraction (fi). The effective dielectric constant for such a case is

given by the expression ∑i

fi(εi − εeff)

(εi + 2εeff)= 0. (8.39)

The value of εeff is derived from the solution of the above equations. How-

ever there may be some convergence problem if there is a wide variation in

optical constants. Maxwell-Garnett theory is more popular and is gener-

ally being used. The applicability and accuracy of the above relations has

to be tested based on the comparison between the computed values and

those of laboratory measurements of two component mixtures. The lab-

oratory measured dielectric constants of several two component mixtures

are in agreement with the effective dielectric constant values derived from

Eq. (8.38). Therefore the M-G rule can be used to derive the average optical

constants of a two component medium. It is possible to extend the method

for a multicomponent mixture as well as for the shape distribution for the

inclusions, which of course introduces several additional parameters.

The dust in comets are more complicated than the simple models consid-

ered in the theoretical treatments as summarized so far. For a real situation

it is necessary to specify several parameters for the aggregate like the size,

shape and the refractive index of the material and for the inclusions, the

degree of porosity, size, shape, location and their orientation. To consider

so many parameters in the actual calculation of cross-section becomes dif-

ficult from the practical point of view as well as computing time required.

In spite of this, in recent times DDA and other methods have been used

extensively in cometary studies. The codes for some of these methods are

available.

Problems

1. Show that for the condition (2πa/λ)<∼1 and with refractive index m =

n− ik, Qabs is given by

Qabs =a

λ

48πnk

(n2 − k2 + 2) + 4n2k2.

2. Express the above relation for Qabs in terms of the dielectric function

ε1 and ε2.

3. Calculate Qsca and Qabs for ice particles of m = 1.3−0.05i, a = 2×10−5

cm and λ = 5 µm.



4. For a particle size distribution of the form n(a)αa−3 for a1 < a < a2,

calculate the constant of proportionality such that the total number is

unity.

5. Assuming m=1, a λ and the size distribution of Problem 3, get an

expression for the total Qabs.

6. Discuss the physical reason for the occurrence of Kinks in the extinc-

tion curve. Why does it smooth out when integrated over the size

distribution function?

References

The theory of scattering from spheres can be found in these classic works:

1. Debye, P. 1909. Ann. Physik, 30, 59.

2. Hulst, Van de 1957. Light Scattering by Small Particles. New York:

John Wiley and Sons.

3. Mie, G. 1908. Ann. Physik, 25, 377.

A discussion of albedos can be found in

4. Hanner, M.S., Giese, R.H., Weiss, K. and Zerull, R. 1981. Astr. Ap.,

104, 42.

The theory of core-mantle particles is worked out in

5. Guttler, A. 1952. Ann. Phys., Lpz. 6 Folge, Bd. 11,5.

The case of core-mantle and cylindrical particles is included in

6. Wickramasinghe, N.C. 1973. Light Scattering Functions for Small par-

ticles, London: Adam Hilger Limited.

A good account is also given in the book

7. Bohren, C.F. and Huffman, D.R. 1983. Absorption and Scattering of

Light by Small particles. John Wiley, New York.

The idea of Discrete-Dipole approximation was proposed in the following

paper.

8. Purcell, E.M. and Pennypacker, C.R. 1973. Ap. J. 186, 705.

More recent work can be found in the following papers.

9. Draine, B.T. and Goodman, J.J. 1993. Ap. J. 405, 685.

10. Hage, J.I. and Greenberg, J.M. 1990. Ap. J. 361, 251.

11. Wolff, M.J., Clayton, G.C., Martin, P.G. and Schulte-Ladbuck, R.E.

1994. Ap. J. (Letters), 423, 51.

12. Wolff, M.J., Clayton, G.C. and Gibson, S.J. 1998. Ap.J., 503, 815.

13. Kimura, H. Kolokolova, L. and Mann, I. 2006. Astron. Astrophys.,

449, 1243.



14. Kolokolova, L., Hanner, M.S. and Levasseur - Regourd, A. -Ch. 2005.

Comets II, eds. M.C. Festou, H.U. Keller and M.A. Weaver, Univ. Ari-

zona Press, Tucson, p. 577.

Maxwell-Garnett and Bruggeman rules are discussed in

15. Bohren and Huffman 1983. Absorption and Scattering of Light by Small

particles, John Wiley, New York.


CHAPTER 9

The Nature of Dust Particles

As was pointed out in the previous chapter, when radiation is incident

on a dust particle (also called the grain), it can scatter as well as polarize

it. The absorbed energy by the dust particles can also be radiated in

the infrared region and this has been seen in comets. There may be also

spectral features in the infrared region depending on the composition of the

dust particle. The radiation pressure acting on the particle can give rise to

dynamical effects as discussed in Chap. 7. Each of these physical effects

could be used to infer the nature and the composition of the cometary

grains. The in situ measurements of Comets Halley, Wild 2 and Tempel 1

performed by the instruments on board the spacecrafts have given valuable

information about the nature and the composition of dust particles. The

laboratory studies of dust samples collected from Comet Wild 2 has revealed

for the first time the true nature of the cometary dust particles. While

it is known that comets do exhibit a great deal of variability, there still

exists a general overall pattern of behaviour in their observed properties.

The emphasis here is on this general pattern of the observed properties of

comets. Here we would like to consider some of these effects.

9.1. Visible Continuum

Comets in general posses a continuum in the visible region of the spec-

trum. The strength of the continuum varies from comet to comet and with

the heliocentric distance. The observed continuum is attributed to the

scattering of the solar radiation by the dust particles. Therefore, the dusty

comets should have a strong continuum. What is of interest, of course,

is the dependence of the continuum as a function of the wavelength. The

usual method adopted for getting the variation of continuum with wave-

249



length is to compare the observed energy distribution of a comet with the

energy distribution of the Sun or a similar star. But the cometary spec-

trum in the visible region is dominated by the emission features from the

molecules. Therefore, the continuum has to be corrected for these emission

features or a spectral region has to be selected where the emission features

are absent or minimal. From these observations, it is possible to determine

the relative continuum intensities of comet with respect to the Sun or a star

similar to the Sun. This yields the wavelength dependence of the observed

continuum of the comet. Since the dependence arises due to the scattering

of the dust particles in the comet, the interpretation of these observations

should give information about the nature of the particles.

Let I0 be the intensity of the incident solar radiation and r, ∆ represent

the sun-comet and the comet-earth distances respectively. The intensity of

the scattered radiation at the Earth at an angle θ is given by (Chap. 8).

Iscat(r, ∆, θ, λ) =I0(λ)

2k2r2∆2[i1(λ, θ) + i2(λ, θ)] (9.1)

where k = 2π/λ.This has to be integrated in general with a size dis-

tribution function. Therefore, the problem reduces to the calculation

of Iscat(r,∆, θ, λ) for a given size distribution function and for the as-

sumed grain properties represented by the refractive index of the material,

m = n − ik. The mean scattering intensities can be calculated from the

Mie Theory as discussed in the previous chapter. Therefore, the expected

intensity distributions can be calculated for various input parameters which

could then be compared with the observations.

There exists a large number of broad band observations of comets in

the wavelength region from 0.5 to 20 µm. To have an idea of the expected

results from the scattering component of the cometary grains, some of the

comets around a few heliocentric distance values and covering various scat-

tering angles can be grouped together. Since the main interest is in the

general shape of the wavelength dependence of the observed fluxes and not

in the absolute amount, the shape of the curves could be shifted to obtain

a mean dependence of the observed fluxes. This procedure enables one to

extend the wavelength range of the scattered radiation for comets up to

λ ≈ 4 µm as can be seen from Fig. 9.1. There exists a close relation in

the various observations. The shape of the observed solar flux variation

with the wavelength is nearly the same as that of the observed shape. This

shows that it is rather difficult to extract the properties of dust particles in

comets from the study of broad band-pass observations. This is borne out


The Nature of Dust Particles 251

Fig. 9.1 Superposed plot of relative observed fluxes of eight comets as a function

of wavelength for heliocentric distances r grouped around 0.3 AU(), 0.7 AU(4),

1.0 AU(0), 2.5 AU(X) and 3.75 AU(•). They cover the scattering angles from around 30to 177. The continuous curve shows the shape of the observed solar flux and the dashed

curve is for Mie calculations with m = 1.38 − 0.039i and N(a) α a−3.5. Observational

data: Bradfield (1974 III) and Kohoutek (1973 XII) (Ney E.P. 1974 Icarus 23, 551);Bradfield (1980 XV) (Ney, E.P. 1982. In Comets, ed. L.L. Wilkening, Univ. Arizona

Press, Tucson, p 323); P/Crommelin (1984 IV) (Hanner, et al. 1985. Astr Ap 152, 177);

P/Giacobini-Zinner (1985 XIII) (Hanner, M.S. et al. 1992. AJ, 104, 386); Halley (1986III) (Bouchet, P. et al. 1987. Astr. Ap. 174, 288; Tokunaga et al. 1988, AJ, 96, 1971);West (1976 VI) (Ney, E.P. and Merrill, K.M. 1976. Science 194, 1051); Wilson (1987VII) (Hanner, M.S. and Newburn, R.L. 1989. AJ, 97, 254).

from the extensive studies of near infrared colour like J-H and H-K, which

show that they cannot be used as a distinguishing feature for the study

of the properties of cometary grains. It also indicates that in general, the

results expected from the scattering effects of grains are not drastic.

There have been several refined studies of the continuum measurements



of comets in the visible region. However, the results are somewhat confus-

ing. In some comets, the observed wavelength distribution of the scattered

radiation has almost the same dependence as that of the solar radiation,

while for others, it produces a large reddening. This arises due to several

factors. As mentioned earlier, the wavelength dependence of the scattered

radiation is derived from observations by comparing the observed contin-

uum with the energy distribution of the Sun. In this procedure, it is very

essential to take into account properly the contribution of the molecular

emissions, before the dust scattered component can be extracted. This is a

difficult and a major problem in scattering studies, as the molecular bands

are quite strong and dominate in the visual spectral region. This leaves

very few clear windows or spectral regions where the observations could

be carried out. However, the situation improves moving towards the red

region. It may also be noted that the continuum becomes stronger as the

dust to gas ratio increases and the emission bands become weaker as the

heliocentric distance of the comet increases. The other difficulty associ-

ated with the early observations is connected with poor spectral resolution.

A combination of these difficulties and the poor response of the detectors

could have been responsible for the differing results of earlier studies. In

recent years it has been possible to overcome some of these difficulties and

it is now possible to get somewhat reliable and consistent results.

From the observational point of view, it is convenient to describe the

scattered radiation or colour from the observed spectra in terms of a quan-

tity called ‘reflectivity’, which is merely the ratio of the flux F (λ) to the

solar flux F(λ)at the same wavelength, i.e.,

S(λ) =F (λ)

F(λ). (9.2)

To be consistent with the cometary spectra, the solar spectrum has to be

made smooth to the instrumental resolution of the cometary spectra before

the ratio is determined. The normalized reflectivity gradient is defined as

S′(λ1, λ2) =

(ds

dλ

)/S. (9.3)

Here (ds/dλ) represents the reflectivity gradient in the wavelength interval

between λ1 and λ2, and S is the mean reflectivity in the same wavelength

region defined as

S = N−1∑

Si(λ). (9.4)

The reflectivity gradient S′(λ1, λ2) gives a measure of the colour. If

S′(λ1, λ2) > 0, it indicates grain reddening.



The reddening curve, in principle, can be extended to the ultraviolet

and infrared wavelength regions by combining various observations. How-

ever, the main difficulty is that since the observations are performed with

different instruments, different wavelength regions etc., it is hard to connect

them with one another. An attempt to combine the data obtained from

the IUE and ground-based observations, along with infrared observations

of Comets Bowell, Stephen-Oterma and Cernis, has succeeded in getting

the reddening curve for the coma in the wavelength region λ = 0.26 to

2.5µm. This showed a gradual reddening with an increase in wavelength.

In more recent times, the reflectivity and the reflectivity gradient have been

determined in a systematic manner from the study of several comets in the

spectral region from 0.5 to around 3 µm. This broad coverage in wavelength

came about as a result of the combination of the scanner observations in

the wavelength region of 0.35 to 0.7 µm and the near infrared observations.

In these studies, several wavelength regions in the cometary spectra, as far

as possible, free of molecular emissions, were selected. This criterion is par-

ticularly difficult in the continuum measurements in the ultraviolet region,

where they are susceptible to emission lines in the spectral region. The de-

rived average behaviour of S′ based on a large number of comets is shown

in Fig. 9.2. The results for Comet Halley also show a similar behaviour of

the reflectivity gradient with wavelength. These results clearly show that

the average S′ decreases systematically from ultraviolet into the near in-

frared wavelength region. In particular, it shows a strong reddening in the

ultraviolet region which decreases slowly with an increase in wavelength.

The neutral point being around λ ∼ 2 µm and the blue beyond λ ∼ 3 µm.

Therefore, comets in general, appear to show an average behaviour for the

reddening as shown in Fig. 9.2, although there are some exceptions. In

addition, there could be spatial as well as temporal variation in colour in

the coma.

Colours of dust grains can also be determined from the difference in

the values of A(θ)fρ (Sec. 9.1.4) between the two continuum filters used

in photometric measurements. Observations of several comets indicate the

presence of reddening.

It is also possible to determine the distribution of surface brightness of

the continuum radiation. The results based on several comets show that

the brightness varies with the nucleocentric distance up to about 105 km

or so, with an average power index of around −1. The particles generated

near the nucleus, as they expand outwards with a constant velocity, will

give rise to a surface brightness variation with an index of −1. The close



Fig. 9.2 The observed normalized reflectivity gradient S′(λ1, λ2) between wavelengths

λ1 and λ2 expressed in percent per 103A is shown as a function of the wavelength for

more than a dozen comets. The horizontal line represents the observational value for eachcomet over the interval between λ1 and λ2. The filled circle represents the mean value

of S′ with each measured wavelength interval. The dashed line is for neutral scattering

with S′(λ1, λ2) = 0. Continuum colours due to scattering changes from red (S′ > 0) toblue (S′ < 0) as the wavelength changes from optical to near-infrared regions (Jewitt,

D.C. and Meech, K.J. 1986. Ap J 310, 937).

agreement between the observed and the expected variation shows that the

fountain model is reasonable. However, with an increase in distance from

the nucleus, the effect of radiation pressure on the grains will start becom-

ing important causing distortion in the observed surface brightness curve.

This is evident from the observed isophotes of comets which show spherical

symmetry close to the nucleus, but distorted at larger distances. Also, the

asymmetry in the coma due to sunward ejection of the dust particles be-

comes prominent. In these situations, the surface brightness in the coma is

also a function of the phase angle. Therefore, the surface brightness profile

of comets can be described reasonably well with the geometrical dilution of

the expanding gas for distances not too far from the nucleus. But, with the

increase in distance, the radiation pressure effect becomes important and

it dominates at large distances causing deviations from the −1 index law.

The interpretation of reddening results in terms of the physical charac-

teristics of the dust grains is complicated, as it involves several parameters,



including the variation in scattering angle, and is also generally based on the

Mie theory of scattering for spheres. Also, in general, it is rather difficult

to entangle the influence of size and the composition of the dust grains.

The early reddening observations of the tails of Comets Arend-Roland

and Mrkos were found to give a fit with model grains with the refractive

index of iron and sizes ∼ 0.3µ. The scattered radiation from the head re-

gions of Comets Arend-Roland and Mrkos could also be fitted with particles

with the refractive indices between 1.25 and 1.50 (dielectric) and the size

distribution proportional to a−4 for the sizes of the particles. The inter-

pretation of later observations of Comets Kohoutek, Bradfield, West and

others seems to indicate the particle sizes to be submicron or micron and

slightly absorbing. The expected intensity distributions have been carried

out for various types of silicate materials as well as for various values for

the constant refractive index (n) with a small absorption part (k). These

comparisons indicate that n ≈ 1.5 to 1.6, k ≤ 0.05 and the particle size

∼ 0.2µ. The observed colour trend of Fig. 9.2 is consistent with dust grain

sizes, a ≥ 1 µm and slightly absorbing.

9.1.1. Albedo

The average value for the albedo of cometary particles can be obtained

from a combination of very simple physical arguments and observations.

The observation required for such a study is the scattered radiation at the

visible wavelength and the total integrated infrared flux from a comet.These

are readily available for many comets. The basic physical idea behind such

a simple method is that the continuum in the visual region arises due to

the scattering process which depends on the scattering efficiency of the

grain. On the other hand the observed infrared radiation is dependent on

the amount of impinging energy absorbed by the grain, which depends on

the absorption efficiency of the grain. For small optical depths and also

neglecting phase dependence effects, the optical surface brightness can be

written as

Sopt(λ) =F(λ)τ

4π=F(λ)Ndlπa

2Qsca(a, λ)

4π. (9.5)

Here Nd is the column density of the dust particles and l is the represen-

tative path length. The infrared surface brightness can be approximated

as

Sir =〈Qabs〉πa2FNdl

4π. (9.6)



Here 〈Qabs〉 represents a mean absorption efficiency for the grain and

F =∫F(λ)dλ, the integrated solar radiation. It is possible to derive

an expression for the average albedo γ of the particles from the above

equations. This is given by

γ

1− γ=FSopt(λ)

SirF(λ). (9.7)

The estimated values of albedo for several comets is ∼ 0.1. In the above

method, the phase dependent scattering is neglected. The derived albedo

for Comet Giacobini-Zinner to around 0.07 to 0.15.

9.1.2. Phase function

The average scattering function or phase function, i.e., the scattered

intensity as a function of the scattering angle, is another important param-

eter in defining the nature of the particle responsible for it. The average

scattering function can be obtained by observing the coma of the comet

as a function of the heliocentric distance or by scanning along the outward

direction of the tail. In such observations the quantity that is changing is

basically the scattering angle.

It is considerably difficult to derive the phase function, since the phys-

ical conditions within the coma change with the scattering angle. The

approximate phase function has been derived from the ratio of the visual

flux, which represents the scattered radiation, to the infrared brightness

that represents the absorbed energy, for the scattering angles between 30

and 150. Since they both come from the same volume of dust, the ratio is

simply proportional to the scattering function of the grains. The derived

phase function for several comets is shown in Fig. 9.3. The phase function

shows a forward scattering lobe for scattering angle ≤ 40 with almost a flat

shape for an angle between 60 and 150. The observation of Comet Halley

carried out upto a scattering angle of 178.63 does not indicate a strong

backscattering peak. The results for Comets Ashbrook-Jackson, Bowell and

Stephen-Oterma also show a similar behaviour of Comet Halley. Therefore,

there appears to be no evidence for the presence of backscattering peak.

The results for Comet Machholz follow the general trend of the continuum

curve for scattering angles less than about 50. The values for Comet Hale-

Bopp is about 50% higher compared to that of Comet Halley for scattering

angles around 140. The nature of the observations is more in conformity

with the expected results from non-spherical particles (Fig. 9.3(b)).



Fig. 9.3 (a) A composite phase diagram of the observed ratio of the reflected to theinfrared flux as a function of the scattering angle derived for several comets. For com-

parison with the data, the curve 3 from (b) is also shown. (b) shows the laboratory data

from spherical (curves 1 and 2) and nonspherical particles (curve 3) (Gehrz, R. D. andNey, E. P. 1992. Icarus, 100, 162).

The phase function for Comet Halley has been determined for phase

angles between 1.5 and 66. This in combination with other results should

give the phase curve over the entire range of phase angles.

The observed phase function is consistent with the expected distribution

of micron size particles with an index of refraction in the ranges 1.3 ≤ n ≤2.0 and k ≥ 0.05.

With regard to albedo, the estimated albedos for different comets have

to be compared at the same scattering angle. The geometric albedo Ap at

λ ∼ 1.2 µm is roughly around 0.025 for scattering angles in the region of

around 150.

9.1.3. Dust production rate from continuum

The production rate of grains can be calculated from a knowledge of

the total number of dust particles in the field of view required to explain

the observed scattered flux at the earth, Fearth. It is given in gm/sec by the



expression

QoptD (r) =

4π∆2Fearth

δ

[M

Eλ

]V (9.8)

where

E =

∫ amax

amin

I0(λ)

2k2

R2r2

[i1(λ, θ) + i2(λ, θ)]n(a)da (9.9)

and

M =

∫ amax

amin

4π3 a

3ρn(a)da∫ amax

aminn(a)da

. (9.10)

Here δ is the linear radius of the projected field of view, ρ is the density of

the grain material and ∆ is the geocentric distance. The mean velocity of

grains is defined by

V =

∫ amax

amina3V (a)n(a)da∫ amax

amina3n(a)da

. (9.11)

In the above expression, n(a)da represents the number of grains of size a in

the size range between a and a+da and V (a) is the velocity of grains of size

a. The summation has to be carried over minimum, amin and maximum,

amax size ranges of the size distribution function. The other symbols have

their usual meanings.

The calculation of dust production rate from the nucleus requires the

knowledge of the density of the grains, the dust particle velocities and the

size distribution of grains. However the density of grains in comets is quite

uncertain. Although the instruments on board the spacecrafts to Comet

Halley were not designed or calibrated to measure the density accurately,

some estimates have been made which are around 0.3 to 1.0 gm/cm3. The

compositional measurements from Giotto and Vega spacecrafts indicate an

average value of ∼ 1 gm/cm3 for the carbon type of particles and a value

∼ 2.5 gm/cm3 for the silicate type particles. The density measurements of

the actual interplanetary dust particles indicate a value ∼ 1 gm/cm3. In

view of the uncertainties, a canonical value of 1 gm/cm3 is generally used

as the production rate roughly scales with the density.

The velocity of the grains has to be obtained from the study of the

dust-gas dynamics in the coma. Either the dust velocities calculated based

on a simplified expression for the dust or by the results based on detailed

dust-gas dynamics may be used.

The size distribution function for cometary dust is generally taken as

n(a)da ∝ aαda (9.12)



where α is a constant. The value of α is generally ∼ −3.5 (Sec. 9.3). Some

variation in the above type of function has also been used.

The production of dust can be calculated from Eq. (9.8), provided the

scattered continuum is known. As discussed earlier, it is rather difficult to

get accurately this quantity due to the presence of strong emission bands

in the visual spectral region. This will also introduce uncertainty in the

derived production rate of grains. However, the difficulty associated with

band emissions improves in the red wavelength region. In addition, the

observed fluxes in the near infrared region of comets essentially refer to the

scattered radiation by the dust particles. Therefore, it may be advantageous

to use the observed fluxes in this spectral region for the calculation of

production rate. Some sample calculated results are given in Table 9.1. It

is interesting to note that the derived production rate of grains from the

observed scattered fluxes and infrared fluxes agree reasonably well.

Table 9.1 Production rate of grains (Kg/s).

Comet Date r(au) ∆ (AU) From IR Fromemission∗ scattered

radiation

Halley 25 Aug 85 (Pre) 2.81 3.16 2.4(4) 3.4(4)

26 Sept 85 (Pre) 2.40 2.18 5.8(4) 5.0(4)

8 Jan 86 (Pre) 0.90 1.29 2.9(6) 2.2(6)2 May 86(Post) 1.65 0.85 5.3(5) 3.8(5)

30 May 86 (Post) 2.05 1.75 2.5(5) 2.4(5)

Kohoutek 10 Dec 73 (Pre) 0.65 1.2 2.0(6) 2.5(6)

16 Dec 73 (Pre) 0.48 1.1 2.0(6) 3.4(6)1 Jan 74 (Post) 0.23 1.0 5.0(6) 8.0(6)

4 Jan 74 (Post) 0.33 0.9 3.3(6) 3.8(6)

∗ Krishna Swamy, K.S. 1991. Astr. Ap. 241, 260.

Pre: Pre-perihelion observation.Post: Post-perihelion observation.

9.1.4. Dust production from A(θ)fρ

In view of the difficulties associated with deriving accurate scattered

radiation of dust from continuum observations as well as its interpreta-

tion, sometimes it may be sufficient to use a quantity which represents the

average property of the grain. This will be particularly useful in the inter-



comparison of observations of various comets. Therefore, a quantity A(θ)fρ

has been used extensively as a proxy dust production in comets and it is

given by

A(θ)fρ =

[2∆r

ρ

]2(Fcom

F

)ρ. (9.13)

Here A(θ), f and ρ represent the average grain albedo for the phase angle

of observation θ, filling factor of the grains in the field of view and the

linear radius of aperture projected on the comet respectively. Fcom is the

observed cometary flux. The filling factor f is given by

f =N(ρ)σ

πρ2(9.14)

where N(ρ) is the number of grains in the field of view and σ is the average

grain cross-section. If the column density N(ρ) ∝ ρ−1, as in the case of the

radial outflow model, the quantity A(θ)fρ becomes independent of the field

of view or the geocentric distance. The advantage of the quantity A(θ)fρ

is that it can be directly determined from the measurable quantities. Since

A(θ)fρ depend upon the phase angle, some correction may be needed on

the measurements. The resulting values of A(θ)fρ which has been corrected

to zero phase angle represents the intrinic production of dust of the comet.

In addition there could be some deviation in the projected density from

ρ−1 variation, which could have some effect on the measurements made

with different aperatures. The quantity Afρ allows comparison of dust

production rates of comets.

The calculated dust production rate A(θ)fρ from observations for

Comet Halley is shown in Fig. 6.6. The resulting variation of Afρ cor-

rected to zero phase angle is quite similar to Fig. 6.6. The power law

constant which are for the pre- and post-perihelion observations give values

of −2.96 and −2.25 respectively. The interesting feature is the pre- and

post perihelion data do not converge near perihelion in contrast to those of

other gas species.

In contrast of Comet Halley, Comet Borrelly has heliocentric distance

dependence for OH of −8.9 which is very steep. Other observed molecules

have also similar steep slopes. These variations are vastly different from

that of A(θ)fρ which has a power law slope of around −3. In addition it

does not show any asymmetry about perihelion.

The dust-to-gas ratio is just the ratio A(θ)fρ/Q(OH). Since the helio-

centric variation of gas and dust is vastly different in Comet Borrelly, the

derived gas-to-dust ratio varies during each apparition. There is a strong



asymmetry for the dust between pre- and post-perihelion data in the case of

Comet Wild 2. The dust-to-gas ratio was found to have a variation of two

orders of magnitude between different comets in the study of a sample of

85 comets. The variation that have been seen among different apparitions

in the same comet could be due to evolutionary effect.

As can be seen from the above discussion, there appears to be a wide

variation in the dust-to-gas ratio in comets and its variation with helio-

centric distance. These could be the consequence of strong seasonal effect

which is also present in the gas species as pointed out earlier.

9.2. Polarization

9.2.1. Linear polarization

The scattered light from cometary dust is usually linearly polarized,

which implies that the electromagnetic wave has preferential plane of oscil-

lation. For randomly oriented particles the electromagnetic wave oscillates

either perpendicular (called positive polarization) or parallel (called nega-

tive polarization) to the scattering plane.

The polarization of the scattered radiation is another important obser-

vation which can give information as to the nature and size of the particle.

In fact, it is found that the polarization is quite sensitive to the shape,

structure and sizes of the dust particles.

The appearance of bright Comets Arend-Roland and Mrkos gave a good

opportunity for making the first good polarization measurements. The mea-

sured polarization at λ ∼ 5000 A and for the phase angle of ∼ 100 for both

the comets is about 20%. Until the recent measurements on Comet Halley,

there were not many systematic continuum polarization measurements on

a single comet over an extended range of heliocentric distances and/or for

scattering angles from 0 to 180. The available observations were referred

either to the fixed scattering angles or to fixed heliocentric distances. The

observations were also referred to mostly in the visual spectral region based

on the use of broad-band passes. Therefore, as in the case of continuum

measurements, the possible contribution of the molecular emission features

has to be taken into account. The observed variations of the polarization

with scattering angle for several comets are shown in Fig. 9.4. They show

a well defined mean polarization curve as a function of the scattering an-

gle. The polarization shows negative values for scattering angles ≥ 160,

a gradual increase with a decrease in scattering angle reaching maximum



Fig. 9.4 The observed dust polarization in several comets denoted by various symbolsis plotted as a function of the scattering angle. The existence of negative polarization

for scattering angles around 170 can be seen. The observations clearly delineate a well

defined polarization curve as a function of the scattering angles. The dark vertical barsare the calculated values (Krishna Swamy, K. S. and Shah, G. A. 1988. Monthly Notices,

233, 573).

polarization value ∼ 20 to 30% around θ ∼ 90 and then decrease for θ’s

< 90. The observed negative polarization is typically around −2% and is

parallel to the scattering plane. The neutral point is around the scattering

angle of ∼ 160. The polarization becomes perpendicular to the scattering

plane for scattering angle less than 160. The extensive polarization mea-

surements carried out in Comet Halley from the visible to the near-infrared

wavelength region show a very similar behaviour as shown in Fig 9.4. This

general behaviour could be taken as a representative of quite conditions

of the coma. There could be a variation of polarization with the level of

activity of the nucleus, such as the presence of jets, bursts or in active

comets.

For interpreting polarization observations, Mie theory has been used

extensively. From a comparison of the observed polarization with those

of calculations, it is possible to restrict the ranges in the input parame-

ters for the dust particles. In general, the smaller size particles have a



complicated polarization behaviour with respect to the wavelength and the

scattering angles. In order to reproduce the observed value of polarization,

one requires the complex part of the refractive index to be small. This is

also consistent with the result that a high value of the complex part can-

not produce negative polarization for large scattering angles. Therefore,

roughly one can put a range for the parameters n and k of 1.3 < n < 2.0

and 0.01 < k < 0.1 and particle size ≥ 1µ.

The usual procedure adopted is to calculate the polarization as a func-

tion of the scattering angle for a grid of real and imaginary part of the

refractive index, particle size distribution and then to try to get the best

fit to the observations. Such an analysis was carried out for Comet Halley

based on the size distribution of grains derived from Vega measurements.

This leads to an average refractive index of the grain for the visible region,

n = 1.39 ± 0.01 and k = 0.035 ± 0.004. The observed linear polarization

with the scattering angle for several comets is also consistent with such a

refractive index (Fig. 9.4). The calculations for rough particles, including

both silicate and graphite grains, also give a good fit to the Comet Halley

polarization measurements.

So far the discussion pertained to the average behaviour of the observed

polarization in various comets. However the polarization data seems to in-

dicate a wider dispersion in polarization values for scattering angle less than

150o. It seems to fall into two distinct classes corresponding to the maxi-

mum polarization. This is shown in Fig. 9.5. One group such as Comets

West, Halley, Levy 1990XX etc. corresponds to comets with high polar-

ization of around 25–30%. The other group of comets such as Kobayashi-

Berger-Milon, Austin etc. appears to belong to low polarization values

around 10–15%. The polarization observations of the bright Comet Hale-

Bopp limited to scattering angles > 132 show distinctly higher values than

the above two groups. Efforts are being made to understand the real nature

of these two apparently two classes of comets.

The polarization also shows wavelength dependence. The wavelength

dependence of polarization has been carried out on several comets. The

characteristic feature is that the polarization increases with increase in

wavelength from the visible region. It also depends upon the scattering

angle. This was noticed in Comet Halley and later confirmed based on

other comets. However few comets have shown that the degree of polar-

ization decreases with increase in wavelength as was observed in Comet

Giacobini-Zinner. The polarization changed from 8% at 4430 A to 5% at

6420A observed at phase angle 44. This type of behaviour was also seen



Fig. 9.5 The observed polarization of comets for phase angles 0.3 to 120. They seem to

divide comets into classes depending on the maximum polarization (Levasseur-Regourd,

A.-Ch, Hadamcik, E. and Renard, J.B. 1996. Astron. Astrophys., 313, 327).

in Comet Halley based on Giotto measurements. This has also been seen

in Comet Tempel 1. The reason for the increase or decrease in polarization

with wavelength is basically related to the sizes of the dust particles in

relation to the wavelength of the observation. This is due to the fact that

the dust particles are efficient scatters when the size of the dust particles

is of the order of wavelength of the incident radiation. Therefore the po-

larization dependence depend upon the size of the dust particles at a given

wavelength.

An extensive polarimetric measurements of the tail of Comet Ikeya-

Seki carried out in 1965 when the comet was at r ≈ 0.3 AU showed the

surprising result that the polarization value changed from +0.20 to −0.42%

with scattering angles from 116 to 136 respectively, for λ = 0.53 µm.

The neutral point (zero polarization) was found to be around a scattering

angle of 125. The linear polarization measurements carried out along the

tail of Comet Halley in April 1986, at r = 1.3 AU showed that they are

very similar at 1000 km, 3000 km from the nucleus and for the envelope

around the nucleus, for the scattering angles between 125 to 160. These



observations showed that there was no change in the crossover angle of linear

polarization and there was also no drastic modification of the grains as they

were transported from the nucleus out to distances of around 1.6×106 km.

The expected polarization calculated as a function of the particle size

shows that it is possible to produce positive and negative polarizations

for grain composition of silicate type and not for that of dirty ice (ice

with a small complex part), graphite or iron. Figure 9.6 shows the typical

result obtained for grain composition of olivine. As the polarization is very

Fig. 9.6 Expected polarization versus particle sizes for λ= 5300 A and for two scattering

angles of 116 and 135. The oscillations will smoothen out with the use of a size

distribution function (Krishna Swamy, K. S. 1978. Astrophys Space Sci. 57, 491).

sensitive to particle sizes, it is possible to get reversal in polarization, if

there is a variation of particle size along the tail of the comet.

At the present time, there exist several kinds of observations pertaining

to scattering and polarization properties of cometary dust. Any proposed

cometary dust model should be able to explain simultaneously all these

observations. Attempts are being made to develop such a model. One

such model proposed to explain all the observed characteristics in the vi-

sual spectral region uses aggregates of 10 to 1000 monomers with sizes

∼ 0.1 µm composed of silicate, carbonaceous material and iron-bearing

sulphides with volume fraction of one-third silicates, two-thirds carbona-



ceous material and a small amount of iron-bearing sulphides. The resulting

mean refractive index derived for this mixture based on Maxwell-Garnet

mixing rule and consistent with the Comet Halley’s in situ abundances is

mav = 1.88 + 0.47i for λ = 4500 A and mav = 1.98 + 0.48i at λ = 6000 A.

The result based on this model can reproduce the correct behaviour of the

observed characteristics of comets such as, the observed shape of anugu-

lar dependence of scattering intensities and polarization. The model can

account for all the observed trends and also the mineral composition of

cometary dust is consistent with the elemental abundances derived from

other considerations. This has led to consideration of more realistic model

which can explain the observations in a quantitative manner.

9.2.2. Circular polarization

So far, the discussion pertained to linear polarization. It is also of in-

terest to see whether circular polarization can be detected in comets as it

can give additional information about the grains. The expected circular

polarization is of the order of 0.5% for certain values of scattering angles

which is quite small compared to the linear polarization. Circular polar-

ization was first detected in Comet Halley with an average value of around

1.7 × 10−3. This was followed by Comet Hale-Bopp which had a circular

polarization of −0.26% in 1997. Since then circular polarization has been

measured in Comets D/1999 S4(LINEAR), C/2001 Q4(NEAT) etc. The

circular polarization seen from comets show left circularly polarized light.

This can be seen from Fig. 9.7 where the phase dependence of circular

polarization is shown for several comets. The figure also show the increase

in circular polarization (absolute value) with phase angle.

The circular polarization is produced only in the case when an inci-

dent unpolarized light after scattering by an ensemble of particles violating

mirror symmetry. This can arise due to the intrinsic property of the parti-

cles itself or it could be due to an external cause. The external processes

could due to multiple scattering in an anisotropic medium or scattering

by aligned non-spherical particles. Scattering by optically active (Chiral)

particles belongs to the other class where the property is intrinsic to the

particle.

For multiple scattering to be the cause of the observed circular polariza-

tion, the optical depth must be appreciable. The estimated optical depth in

a cometary atmosphere appears to be reasonable and therefore in principle

this mechanism can account for the observed circular polarization.



Fig. 9.7 The variation of circular polarization with phase angle for Comets Halley,

Hale-Bopp, LINEAR and NEAT (Rosenbush, V., Kiselev, N. and Kolokolova, L. 2008.Organic Matter in Space, eds, Kwok S. and Sandford S., Cambridge Univ. Press, Cam-

bridge, p. 311).

In the case of scattering by aligned non-spherical particles, the main

problem is associated with the mechanism of alignment of the particles.

The other class of particles is where the circular polarization arises from

scattering of light by dust particles composed of optically active material

(Chiral). The property of such a material is that they have different refrac-

tive indices for left and right circularly polarized light which automatically

separate the two polarizations.

It is interesting to note that the above effect is quite strong in complex

organic molecules. This is due to the fact that they exist in two forms called

Left-handed(L) and Right-handed(D). They arise mainly because the or-

ganic molecules are optically active as well as possess circular dichroism.

The biomolecules on the earth are only L-amino acids and D(sugar). This

is also found to be case in the case of Murchison meteorite (an extrater-



restrial source). Therefore the dust particles present in comets must be

Chiral organic in character. This is consistent with the left-handed circular

polarization seen from comets. The model based results are in reasonable

agreement with the observations.

These studies are highly relevant, as comets are believed to have trans-

ported organic molecules on to the earth by cometary impacts (Sec. 13.6).

It is intrigueing that the biomolecules present on the earth, L-amino acids,

could actually be the ones that were transported from comets during earlier

times.

9.3. Grain Sizes

Dust particle detectors were used on Giotto and Vega missions to Comet

Halley to record dust impact rate and thus to measure the mass distribution

of particles. The instrument from both the missions covered together the

mass range of around 10−6 to 10−16gm. The size distribution of particles

can therefore be inferred from these mass measurements. The cumulative

mass distribution function for the observed data is consistent with α =

−3.7(n(a) ∝ aα). The measured mass distribution of dust particles for

particles masses > 10−6gm is found to be much flatter compared to smaller

masses. This indicated that the total mass is mostly contributed by large

size particles. The in situ mass distribution of dust particles in the coma of

Comet Wild 2 was also carried out by the stardust misssion. The results of

these measurements was found to be similar to that of Comet Halley. The

in situ measurements also showed the dominance of small size particles,

a ≤ 0.1 µm, which cannot generally be detected through observations made

in the visible region.

It is possible to deduce the size distribution function of the dust particles

from the model fitting to the observed dust tail of comets. These studies

indicate the size distribution function is characterized by the power law

index of −3.5. The analysis of the nucleus of Comet Tempel 1 acquired

with camera aboard Rosetta spacecraft in the orange filter 6450 A with

the Monte Carlo approach gave an index of −3.0 for the power law size

distribution function.

Large size particles are also present in comets. This comes from several

observations. The absence of 10µm silicate emission feature in the anti-

tail of Comet Kohoutek puts a lower limit to the particle size, a ≥ 5 µm

(Sec. 9.4.2). This conclusion is also consistent with the results based on



dynamical considerations of the anti-tail which gives for the particle sizes,

a ≥ 15 µm. Supportive evidence for the presence of large size dust particles

comes from the study of meteor showers which are associated with comets.

In addition dust trails seen in comets require millimetre to centimetre size

dust particles.

The dust particles emitted from comets span a wide range in sizes, from

submillimetre to millimetre to centimetre. The derived size distribution

of dust particles based on various studies is a representative value for all

comets, but it could vary from comet to comet and can also vary within

the coma. The variation in the size distribution of dust particles with

distance in the coma can arise as a resultant effect of sorting of particles

due to radiation pressure effect and sublimation of volatiles. The power

law index, α ≈ −3.5 of the size distribution function, is also characteristic

of intersteller grains and is typical of collisionally evolved bodies.

9.4. Infrared Measurements

There are very few windows in the infrared region that are available for

ground based observations, as most of the infrared radiation is absorbed

by the Earth’s atmosphere (Fig. 9.8). Therefore, most of the observations

are carried out with broad band passes. Several such systems exist which

have been used by different observers with slight variations in their mean

wavelengths and band widths. (Table 9.2).

Table 9.2 Infrared band passes and their wavelengths.

Arizona System IRTF SystemBand pass Wavelength ∆λ Wavelength ∆λ

(µm) (µm) (µm) (µm)

J 1.26 0.20 1.20 0.3

H 1.60 0.36 1.60 0.3K 2.22 0.52 2.2 0.4L 3.54 0.97 3.55 1.05

L′ - - 3.78 0.57M 4.80 0.60 4.7 0.57N 10.6 5.0 10.50 5.0

Q 21.0 11.0 20.6 9.0

Adapted from Hanner, M.S. and Tokunaga, A.T., 1991. InComets in the Post-Halley Era, eds. R.L. Newburn, Jr et al.

Kluwer Academic Publishers, p. 70.



Fig. 9.8 The atmospheric transmission is plotted as a function of wavelength for the

region of 1 to 20 µm. The spectral band passes corresponding to J to Q are also shown(Courtesy Charlie Lindsey).

The first infrared observations made on Comet Ikeya-Seki in 1965 in the

wavelength region of 1 to 10 µm showed clearly that the comet was very

bright in the infrared wavelength region and its colour temperature was

higher than that of a black body at the same heliocentric distance. These

results have been confirmed based on the infrared observations of many

more comets. Most of the observations on comets before Comet Halley

were limited to broad band infrared observations in the spectral region

around 2 to 20 µm. However, for Comet Halley it has been possible to

get very good broad band and spectroscopic data in the middle infrared

and far-infrared wavelength regions based on ground based, airborne and

spaceborne instrumentation.

The infrared observations of comets can provide another independent

method for extracting significant information on the physical nature of the

cometary grains. This is due to the fact that the observed infrared radiation

arises from the re-radiation of the absorbed energy by the dust particles



which in turn depends on the nature and composition of the dust. From a

detailed comparison of the cometary infrared radiation with the expected

infrared fluxes based on grain models, it is possible to infer the physical

and the chemical nature of cometary grains. The grains are considered to

be in radiative equilibrium with the incident solar energy. The equilibrium

temperature of the grain is, therefore, determined by a balance between

the absorbed radiation which is mostly in the ultraviolet and visible regions

and the emitted radiation which is in the far infrared region. This can be

expressed by the condition

Fabs(a) = Fem(a, Tg) (9.15)

where

Fabs(a) =

(Rr

)2 ∫F(λ) Qabs(a, λ)πa2dλ (9.16)

and

Fem(a, Tg) =

∫πB(λ, Tg)Qabs(a, λ)4πa2dλ (9.17)

where F(λ) representing the incident solar radiation field at wavelength λ,

Qabs(a, λ) is the absorption efficiency and B(λ, Tg) is the Planck function

corresponding to the grain temperature Tg. The difference in the geometri-

cal areas between the two sides arises as the grains absorb in one direction

and emit in all the directions. The calculation of the grain temperature

involves a knowledge of the size and composition of the grains. Knowing

the grain temperature, it is possible to calculate the infrared emission from

Eq. (9.17).

In general, the emission has to be integrated over the size distribution

function to get the total infrared emission from the grains. Therefore, the

total infrared emission at the Earth is given by

Fem(λ, r) =1

∆2

∫ α

0

n(a)πa2Qabs(a, λ)B(λ, Tg)da (9.18)

where n(a)da represents the relative number of grains in the size interval

between a and a + da and ∆ is the geocentric distance of the comet. If

there are grains of various types present at the same time, then the total

observed infrared radiation is the sum total over all the grain types, j, i.e.,

Ftotal(λ, r) =

j∑i=1

Fem(λ, r)i xi (9.19)



where xi is the fraction of the grain population of type i. The correspond-

ing mass of grains of type i is given by

Mi =

∫ α

o

4π

3a3ρin(a)da (9.20)

where ρi is the density of the ith type of the material.

The important observation which gave some clue to the possible nature

of the grain was the detection of a broad 10 µm emission feature in Comet

Bennett (C/1969 Y1). Another feature at 20 µm also appears to be present

in many comets (see Fig. 9.9). However, the whole profile does not show up

in these broad band observations as the last band pass is at around 20 µm.

The observed features at 10 and 20 µm are widely believed to be due to a

silicate type of material (Sec. 9.5). The moderate resolution observations of

Comet Halley made in the 10 µm region with Kuiper Airborne Observatory

has given important information about the nature of the silicate material

in showing that it is in crystalline in nature.

Observations of the 10 µm feature in the spectra of comets show that

the apparent strength of this feature varies from comet to comet and also

with the heliocentric distance for the same comet. The strength of the 10

and 20 µm features is a function of the particle size. It becomes weaker with

an increase in size of the particles (Sec. 9.4.2). Therefore, silicates can still

be the dominant component of the grain even if no 10 µm feature is evident

in the spectra. Another new emission feature near 3.4 µm was detected in

the spectra of Comet Halley. This also appears to be a common feature

of most of the observed comets. The 3.4 µm feature is a characteristic of

the C-H stretching vibrations and indicates the presence of some form of

hydrocarbons. Taken together, the infrared spectra of comets suggest, there

are two components to the grains − silicates and some form of carbon.

Several attempts have been made to reproduce the observed infrared

radiation from comets using models based on assumed optical properties

for the cometary dust particles. They include models such as, assuming

constant refractive index for the material or its variation with wavelength

or a mixture of materials such as silicate and some form of carbon with

some size distribution function etc. (See Sec. 9.5.2).

9.4.1. Dust production from infrared observations

The production of dust can also be determined from the observed in-

frared radiation of comets. The observed infrared radiation at the Earth,



Fig. 9.9 Observed infrared fluxes plotted as a function of wavelength for Comet Ko-

houtek. The fitted black body temperataure for each of the curves along with the factor

by which the temperature exceeds the black body temperature is also shown. (Ney, E.P. 1974. Icarus, 23 551).

Fearth is related to the total number of grains in the coma, N(r) through

the relation

Fearth =Eλ

4π∆2(9.21)

where Eλ represents the infrared radiation of the comet given by

Eλ = B(λ, Tg)Qabs(a, λ)4πa2N(r). (9.22)



Here ∆ is the geocentric distance. A rough estimate for the production rate

can be obtained from the relation

QD(r) ' N(r)

τ(9.23)

where τ is the lifetime of the grain. This gives an average dust mass rate of

about 4× 106 gm/sec for the assumed lifetime of the grain τ ' 4× 104 sec .

However, in general, it is necessary to take into account the effect of the size

distribution of grains. Therefore, the production rate of grains in gm/sec,

as in the case of continuum can be calculated from the relation

QiRD (r) =4π∆2Fearth

δ

[M

Eλ

]V (9.24)

where

Eλ =

∫ amax

amin

B(λ, Tg)4πa2Qabs(a, λ)n(a)da. (9.25)

The expressions for M and V are given by Eqs. (9.10) and (9.11). All the

symbols which occur in these equations are described in Sec. 9.1.3 or have

their usual meanings. The calculation of the production rate of grains from

Eq. (9.24) involves the knowledge of density, velocity and size distribution

function for the grains. A discussion regarding these quantities is also

outlined in Sec. 9.1.3. The production rate of grains has been calculated

for various comets. The results for Comet Halley for heliocentric distances

between 2.8 and 0.6 AU show a close relation with an r−4 dependence,

while the results for Comet Kohoutek show a close relation for heliocentric

distances between 0.15 and 1.5 AU with an r−2 dependence. It is interesting

to see whether there exists a general behaviour of the production rate of

dust with the heliocentric distance in comets. This could be carried out

by the superposition of the shape of the derived dust production rate with

the heliocentric distance of various comets and the results are shown in

Fig. 9.10.

The gas (composed of H2O) and the dust coming out of the nucleus of

a comet flow outwards in such a way that the gas carries the dust with it.

Therefore, to a first approximation, the heliocentric variation of the dust

production rate and the water production rate should be very similar as can

be seen from Fig. 9.10. The dependence of r−2 for r ≤ 1 AU arises due to

the solar flux falling of as r−2 and it then deviates from this dependence due

to the effect of temperature dependence of the vapour pressure. The ratio

of gas to dust for a representative sample of comets is given in Table 9.3.



Fig. 9.10 A composite diagram showing the general shape of the variation of the dust

production rate with heliocentric distance is derived from several comets, denoted byvarious symbols. There is a tight relation with heliocentric distance r showing r−2

variation up to r ≤ 1 AU and r−4 for r ≥ 1 AU. The dashed curve shows the shape ofthe derived water production for r0 = 2.4 and 2.6 respectively (Krishna Swamy, K. S.1991. Astron. Astrophys., 241, 260).



Table 9.3 Gas to dust ratio in comets (r = 1 AU).

Comets Q(H2 O) Q(dust) H2O/dust

(Kg/sec) (Kg/sec)

Bennett (1970 II) 1.5(4) 1.5(3) 10

Churyumov-Gerasimenko (1982 VIII) 1.1(2) 5.4(1) 2P/Crommelin (1984 IV) 1.8(2) 1.2(3) 0.1

P/Encke 2.1(2) 2.8(1) 7

P/Giacobini-Zinner (1985 XIII) 1.2(3) 3.8(2) 3P/Grigg- 9.0(1) 2.0(1) 5

Skjellerup (1982 IV)

P/Halley (1986 III) 1.5(4) 5.4(3) 3

IRAS-Araki-Alcock 2.2(2) 2.0(2)a <∼ 11.1(1983 VII)

Kobayashi-Berger-Milon (1975 IX) 6.0(2) 1.0(2) 6

Kohoutek (1973 XII) 9.2(3) 4.6(2) 20Sugano-Saigusa-Fujikawa (1983 V) 8.1(1) 2.0 40

West (1976 VI) 1.5(4) 3.5(3) 4

Wilson (1987 VII) 1.1(4) 1.8(3) 6

(Krishna Swamy, K.S. 1991. Astron. Astrophys., 241, 260).

9.4.2. Anti-tail

The characteristic feature of Comet Kohoutek was the presence of the

anti-tail seen shortly after the perihelion passage. The infrared measure-

ments of this comet made on the same day showed the presence of 10 µm

emission in the coma and in the tail, but not in the anti-tail (Fig. 9.11).

The absence of 10 µm feature in the anti-tail immediately puts a lower limit

to the particle size, as can be seen from Fig. 9.12. The figure shows the

shape of the emission curves for grains of silicate type for different sizes

and for grain temperatures of interest. It shows that the strength of the

10 µm feature is a function of the particle size and for a ≥ 5 µm, the feature

actually disappears. The 10 and 20 µm features in small size grains show

up as they are optically thin. As the size of the particle gets larger and

larger, the material becomes optically thick and the feature gets washed

out. Therefore, the particles present in the anti-tail appear to be of a much

larger size compared to those present in the coma or in the tail region. This

conclusion is also consistent with the results based on dynamical consider-

ations of the anti-tail which gives for the particle sizes, a ≥ 15 µm. The

strength of the silicate feature can change from comet to comet. It can

also change with time for the same comet as was the case for Comet Brad-

field. This comet had the silicate signature in the observations of March



Fig. 9.11 Measured energy distributions in the coma, tail and anti-tail of Comet Ko-

houtek are plotted as a function of wavelength. Observations refer to January 1.7, 1974.

The absence of 10 micron feature in the anti-tail observations can be seen. (Ney, E. P.1974. op. cit.).

21.9, 1974 (r = 0.51 AU and ∆ = 0.73 AU), but was clearly absent in the

observations of April 5.8, 1974 (r = 0.67 AU and ∆ = 0.69 AU). These ob-

servations show that the sizes of the emitted grains (Fig. 9.12) could vary

from comet to comet, as well as with the heliocentric distance for the same

comet. The supporting evidence for the large size particles in the anti-tail

also comes from the colour temperature measurements of Comet Kohoutek.

It showed that the colour temperature for the anti-tail was much cooler and

therefore closer to the black body temperature than that of the coma and

the tail temperature.



Fig. 9.12 Shape of emission curves for different sizes, grain temperatures and for ma-terial of moon samples are shown. (a) For moon sample 12009, Tg = 550K and sizes of

0.2, 1.0 and 5.0 microns. (b) For moon sample 14321, Tg = 400 and 700 K. Long-dashed,

continuous and dashed curves refer to particle sizes of 2, 5 and 10 microns respectively.(Krishna Swamy, K. S. and Donn, B. 1979. A.J., 84, 692).

9.5. Spectral Feature

The most direct way of getting information about the composition of the

grain is through the detection and identification of characteristic spectral

features such as the vibration-rotation bands of molecules. It is important

to note that such bands retain their identity even when the molecules are

in the solid state.



9.5.1. Silicate signature

As already remarked, the clue to the possible chemical composition of

the cometary grain came from the infrared measurements themselves, with

the first detection of a broad emission feature at 10µm in Comet Bennett.

This was confirmed by the observation of many other comets, although the

strength may vary from comet to comet with heliocentric distance.

The next logical step is to try to identify the nature of the material

responsible for giving rise to this feature. Laboratory measurements of

the absorption spectra of various types of rocks and minerals carried out at

room temperature or at liquid nitrogen temperature (89K) have a common

property in showing strong and broad absorption feature around 10 µm

(Fig. 9.13). This is attributed to Si-O stretching vibrations. From the

similarity between the laboratory spectra and the cometary spectra, it is

suggested that the 10 µm feature observed in cometary spectra is due to

some type of silicate material. There is the other broad feature present

in the laboratory spectra around 20 micron arising out of Si-O-Si bending

vibrations. This feature has also been seen in cometary spectra confirming

the silicate nature of dust. The laboratory spectra in addition to showing

the main features at 10 and 20 µm, also show a few other secondary features

which are the characteristics of the particular types of silicate material. If

these features show up, then it could be used as a diagnostic of the dust

composition. The silicate material also shows another feature around 30 to

33 µm as was observed in the laboratory absorption spectra of γ− Ca2SiO4,

Olivine (Mg2SiO4) or other silicates. Unlike the 10 and 20 µm features, the

feature around 30 µm seems to depend more on the chemical composition

of the material.

The observed shape of the 10 µm feature in comets should be able to

provide information about the composition of the grains, since different sil-

icate materials have different band profiles depending upon Si-O stretching

frequencies. The observations carried out on Comet Halley with the Kuiper

Airborne Observatory and ground-based telescopes clearly showed a well

defined spectral feature at 11.2 µm located inside the broad 10 µm emis-

sion feature (See Fig. 9.14). This feature is attributed to crystalline olivine

[(Mg, Fe)2 SiO4] based on good spectral match with the observed spectral

emissivity of Mg-rich olivine. The strong 10 µm emission feature has been

seen from a number of comets. Comet Hale-Bopp had the strongest silicate

emission among all the observed comets. It was also unusual in showing

a strong silicate feature even at 4.6 AU pre-perihelion. A typical 10 µm



Fig. 9.13 Absorption spectra of Orthopyroxene ionosilicates. Sample curves shown (in

cm−1) are for the enstatite-hypersthene series of En62, En75 and En84. (Adapted fromLyon. R. J. P. 1963. NASA Technical Note, NASA TN D-1871).

spectral feature observed in a comet is shown in Fig. 9.14.

There are three strong featurers at 9.2, 10.0 and 11.2 µm. The other fea-

tures which are minor in nature is at 10.5 and 11.9 µm. The 11.2 µm feature

is due to crystalline olivine, as was seen earlier, in the Kuiper Airborne Ob-

servations on Comet Halley. Crystalline olivine has also a secondary peak

at 10 µm and a weak feature at 11.9 µm. The broader 10 µm maximum in

the cometary spectra is characteristic of amorphous olivine. The charac-

teristic signature of pyroxene [(Mg,Fe)SiO3] is the presence of a feature at

9.2 µm. Therefore the cometary 9.2 µm feature corresponds to amorphous

Mg-rich pyroxene. Crystalline pyroxene has also a feature near 9.3 µm.



Fig. 9.14 Shows the shape of the emission feature around 10 µm in Comets Halley (r= 0.79 AU, line) and Hale-Bopp (r = 0.92 AU, dots) with various spectral signatures.

(Hanner, M.S. et. al. 1999. Earth, Moon and Planets, 79, 247).

Crystalline pyroxene in general have a wide range of spectral features. The

peak at 10–11 µm contribute to the width of total feature and the structure

near 10.5 µm.

9.5.2. Mineralogy of dust particles

Comet Halley

The mineralogical composition of dust particles was first obtained for

Comet Halley based on in situ measurements. This came from the studies

of mass spectra of around 5000 particles made with instrument on board

Giotto and Vega Spacecrafts. The mass spectra basically provide the el-

emental composition of the dust particles. The mass spectra clearly indi-

cated the presence of dust particles with rock-forming elements Mg, Si, Ca,

and Fe. It also showed the presence of dust particles containing both rock

and CHON elements (Sec. 9.6).

The next step is to get some information on the mineralogical composi-

tion of the dust particles. From a knowledge of the abundances of various

elements, it is possible to make a systematic analysis of the variation among



the observed grains and make some broad classification of its composition.

For this purpose the cluster analysis method can be used. Cluster analysis

is a stistical method of grouping a set of data points to look for correlations

among them. This method has been applied to classify Comet Halley’s dust

particles. In this study, the abundant inorganic elements Na, Mg, Al, Si, S,

Ca and Fe have been considered. The result of these studies and with the

observed distribution of Fe/(Fe+Mg), Mg-Fe-Si, Mg-Fe-O and Mg-Fe-S, it

has been possible to characterise the particles into a few major mineral

groups. The resulting major mineralogical composition of dust particles

in Comet Halley is given in Table 9.4. As can be seen from the table,

the dominant component is the Mg-Si-O rich particles (olivine, pyroxene

etc.). The other group comprising around 10% are the particles of Fe, Ni

and sulphides. These particles could consist of Pyrrhotite (Fe1−xS) and

Pentlendite(Fe Ni)9 S8. All these results indicate that the cometary dust is

an unequilibrated heterogeneous mixture of minerals containing both high

and low temperature condensates.

Table 9.4 Estimated mineralogical composition of Comet Halley Dust.

Estimated Mineral

Mineral group proportion chemistry possible minerals

Mg silicates > 20% Fe-poor, Ca-poor Mg-rich pyroxene and/or olivine

Fe sulfides ∼ 10% some Ni-rich pyrrhotite, pentlandite

Fe metal 1-2% Ni-poor kamacite

Fe oxide < 1% magnetite

Schulze, H., Kissel, J. and Jessberger, E.K. 1997, From Star Dust to Planetesimals, ASP

Conference Series, Vol. 122, p. 397, eds. Y. Pendleton and A.G.G.M. Tielens, By the kindpermission of the Astronomical Society of the Pacific Conference Series.

Comet Hale-Bopp

The remarkable spectra taken with Infrared Space Observatory in the

wavelength region 5 to 45 µm on Comet Hale-Bopp at heliocentric distance,

r = 2.9 AU showed the true nature of the grain material. The spectra show

clearly five strongest emission features at 10, 19.5, 23.5, 27.5 and 33.5 µm

(Fig. 9.15). The wavelength of all peaks correspond to Mg-rich crystalline

olivine (forsterite, Mg2SiO4) when compared with the laboratory spectra.

Minor features present in the spectra are attributed to crystalline pyroxene



Fig. 9.15 The upper curve shows the silicate emission features in Comet Hale-Bopp

observed with ISO, in the wavelength region 5 to 45 µm. The lower curve shows the

modeled spectrum of forsterite from the laboratory data (Crovisier, J. et al. 1997.Science, 275, 1904).

[(Mg,Fe)SiO3]. The agreement between the two is excellent. The infrared

observations of comets have cearly shown that silicate grains in comets

are Mg-rich. Models have been proposed to match the observed infrared

spectra of Comet Hale-Bopp based on a mixture of silicate minerals. A

good fit could be obtained with the following five emission components:

black body 1(T = 280 K), black body 2(T = 165 K), Mg-rich olivine

(forsterite, cry O1), Ortho pyroxene (cry O-pyr) and amorphous (Am Pyr)

silicates. The relative abundances of silicates are Cry O1:Cry O-pyr: Am

pyr = 0.22:0.08:0.70. The temperature of silicate grains used in the model

is 210 K. Therefore the spectral structure indicate complex mineralogy for

cometary silicates containing both crystalline grains of olivine and pyroxene

as well as amorphous silicates. The observed mineralogy in Comet Hale-

Bopp is similar to pyroxene-rich and olivine-rich chondritic aggregate IDPs

thought to originate from comets. The grains in Comet Hale-Bopp is also

Mg-rich which is also in agreement with the chemical composition in the

coma of Comet Halley.

The ISO spectra of young stars with a dust disk such as Herbig’s Ae/Be

star HD 100546 is strikingly similar to that of Comet Hale-Bopp as can be



seen from Fig. 9.16. This shows the similarity in the mineralogy of the

grain material.

Fig. 9.16 Comparison between the ISO-SWS spectrum of Ae/Be star. HD 100546 (top

curve) with the corresponding spectrum from Comet Hale-Bopp (bottom curve). Thestriking resemblance between the two curves can clearly be seen (Malfait, K., Waelkens,

C., Waters, L.B.F.M. et al. 1998. Astron. Astrophys., 332, L25).

Comet Tempel 1

Deep Impact mission gave the first opportunity to look at the deeper

layer material ejected from Comet Tempel 1. The spectral characteristics

of this material was observed by the Spitzer Space Telescope in the spec-

tral region from 5–35 µm. The emission from the normal coma and the

nucleus, which is representative of the pre-impact spectrum was subtracted

out from the observed spectrum. The resulting emission spectrum repre-

sent basically the emission from the ejected material. This is shown in the

Fig. 9.17. The conspicuous features in the spectrum are carbonates in the

region 6.5 to 7.2 µm, pyroxenes in the region 8 to 10 µm, olivines at 11 µm

and sulphides in the region 27 to 29 µm. The calculated spectral emission

model is based on measured thermal emission spectra of micrometer size



Fig. 9.17 (a) Comparison of the observed spectrum of Comet Tempel 1 ejecta obtained

from Spitzers Space Telescope at r = 1.51 AU with the best fit model using various kindsof materials for the dust particles. (b) The residual spectrum after subtracting the best

fit model (a). The best fit with non-silicate species are overlaid in the figure (Lisse, C.M.

et al. 2007. Icarus, 187, 69).

particles of over 80 mineral species. The particle composition, size distri-

bution and temperatures were determined through search in phase space.

Several checks were applied for the correctness of the model such as, min-

erals detected are the major species seen IDPs, the derived abundances

are consistent with solar system abundances as well as the deduced par-

ticle distribution and temperature agrees with the findings of instruments

on Deep Impact etc. The phase space search eliminated large number of

mineral species from the Comet Tempel 1 ejecta. The best model fit to the



observed spectrum is shown in Fig. 9.17(a). It clearly shows the presence

of compositional signaturers due to rich silicates and phyllosilicates, car-

bonates, water-ice, amorphous carbon, sulphides and polycyclic aromatic

hydrocarbons(PAHs). Figure 9.17b show the residual emission features af-

ter subtracting the best-fit model curve to the observed curve (Fig. 9.17(a)).

The non-silicate species which give a fit to the residual emission spectra is

also shown in Fig. 9.17(b). The list of all the identified features in the emis-

sion spectra of Comet Tempel 1 is given in Table 9.5. The derived relative

atomic abundances of Si, Mg, Fe, Ca and Al in the ejecta dust of Comet

Tempel 1 are similar to the relative abundances in the solar system and C1

chondrites. The emission spectra of Comet Tempel 1 ejecta clearly show

the presence of PAH from the detection of features at 5.2, 5.8, 6.2, 7.7 and

8.2 µm. Therefore the spectral structure present in the infrared spectra of

Comet Tempel 1 has shown the complex mineralogy of these dust particles.

Table 9.5 Dust composition of Comet Tempel 1 ejecta

from model fitting.

Species Molecular Nmoles

Weight (rel.)

Olivines

Amorph olivine(MgFeSiO4) 172 0.35Forsterite(Mg2SiO4) 140 0.70

Fayalite(Fe2SiO4) 204 0.18

PyroxenesAmorph pyroxene(MgFeSi2O6) 232 0.06

Ferrosilite(Fe2Si2O6) 264 0.50

Diopside(CaMgSi2O6) 216 0.18Orthoenstatite(Mg2Si2O6) 200 0.16

Phyllosilicates

Smectite nontronite 496 0.07Na0.33Fe2(Si,Al)4O10(OH)2.3H2O

CarbonatesMagnesite(MgCO3) 84 0.11Siderite(FeCO3) 116 0.17

Metal sulphidesNiningerite(Mg10Fe90S) 84 0.92Water

Water ice(H2O) 18 0.27Water gas(H2O) 18 23.7

‘Organics’

Amorph carbon(C) 12 1.45PAH(C10H14) (178) 0.022

Adapted from Lisse, C.M. et al. 2007. Icarus, 187, 69.



The results of Comet Tempel 1 are consistent with the Comet Halley

flyby results which showed the presence of silicates, water, sulphides and

carbonates.

The same model has been used for the re-analysis of ISO Spectrum in

the region 2.4 to 4.5 µm of Comet Hale-Bopp (r = 2.8 AU) and the young

stellar object HD 100546. The results are qualitatively similar in showing

emission signatures due to silicates, cabonaters, Phyllosilicates, water ice,

amorphous carbon and sulphides. But there are some differences as well.

Comet Wild 2

The Stardust misssion which collected samples of dust particles from

Comet Wild 2 has given interesting results with regard to their mineralogy.

It showed the importance of sulphur and nitrogen chemistry in the dust

particles. It is likely that sulphur atom which is generally associated with

mineral phase result have came from mineral such as troilite.

Infrared studies of these particles has shown the presence of amorphous

silicate. Among crystalline silicates, forsterite, enstatite, olivine and diop-

side with pyroxene has been seen. FeNi metal grains and FeNi sulphides

have been seen. The range of Fe/Mg ratios seen in the particles indicate that

Comet Wild 2 is unequilibrated. The presence of particles with nitrogen-

rich chemistry and lower abundance of O implies that nitriles or poly-

cyanides may have been present. The interesting finding is that dust par-

ticles which are mineralogically similar to meteoritic Calcium-Aluminium-

rich inclusions have been found. Ca-Al-rich dust particles are the oldest

samples in the solar system and contain minerals in abundance.

Deep Impact mission of Comet Tempel 1 provided detailed mineralogical

composition of this comet. Among the various minerals, only forsterite was

seen in both the Comets Tempel 1 and Wild 2. Iron sulphide has been seen

in Comet Wild 2, but not FeMg sulphides. The major difference between

the two comets is the presence of carbonatess and hydrated silicates in

Comet Tempel 1 but not in Wild 2 samples. These species have been

seen from meteorites. Its formation is generally believed to arise from

hydrothermal alteration inside a wet parent body.

The vast difference in the observed mineralogy betwen the two Jupiter

family of Comets Tempel 1 and Wild 2 could be due to several reasons. It is

possible that the two comets themselves are different or it could have been

formed in different regions. In addition the synthesis of observed infrared



spectra of Comet Tempel 1 was carried out with laboratory minerals which

may be different from those present in comets that have gone through

various processes. Since comets were formed from aggregates of materials, it

is quite possible that some comets possess hydrated silicates due to melting

of ice. Lastly Comet Tempel 1 site of impact appeared to show the presence

of impact craters. Therefore it is possible that the hydrated silicates could

have been formed inside Comet Tempel 1 due to the impact. However this

appears not the case for Comet Wild 2.

9.5.3. The C-H stretch feature

A new emission feature near 3.4 µm was first detected by Vega 1 space-

craft in the spectra of Comet Halley. This was confirmed by several ground-

based observations. The laboratory absorption spectra of organic materials

both in gaseous and solid phase generally show a strong feature around

3.4 µm. Most of the C-H stretching vibrations fall in the wavelength re-

gions around 3.1 to 3.7 µm (see Fig. 3.4). Therefore, the observed feature

in Comet Halley is attributed to C-H stretching of some organic material.

The 3.4 µm feature has subsequently been seen from several other comets

indicating the common nature of cometary organics. This feature is also

seen from some IDPs collected in the stratosphere. The intensity of the

3.4 µm feature as observed by Vega spacecraft seemed to vary inversely

with the projected distance from the nucleus in a manner similar to the

parent molecule indicating the possible detection of a new parent molecule.

The apparent profile of 3.4 µm feature in several comets is around

3.36 µm (Fig. 9.18). In addition, there is evidence for the presence of

two weaker features at 3.28 and 3.52 µm. The vibrational frequencies of

CH3 group occur at 3.37 µm and 3.48 µm and that of CH2 group at 3.42 µm

and 3.50 µm. Therefore, the observed features in comets appear to indicate

the presence of spectral signatures of − CH3 and − CH2 functional groups.

The structure, width and the intensity of the observed 3.4 µm emission

feature suggest that several of these species may be present.

The feature at 3.52 µm is attributed to ν3 band of Methanol (CH3OH).

Other rotational-vibrational lines of methanol occurring at 3.33 µm(ν2) and

3.37 µm(ν9) contribute to the broad 3.36 µm emission. Detailed modelling

with methanol indicate a residual feature. This could come either from

other weaker lines of methanol, for which the required data is not available

or it could come from some other species.

The other feature present at 3.28 µm is characteristic of aromatic com-



Fig. 9.18 The spectral feature around 3.4 µm region observed from Comet 109P/Swift-Tuttle at r = 1.0 AU (Disanti, M.A. et al. 1995. Icarus, 116, 1).

pounds(PAHs). The feature seen at 3.29 µm from interstellar matter is gen-

erally associated with other features occurring at 6.2, 7.7, 8.6 and 11.3 µm.

Some of these features have also been seen in the infrared observations of

Comet Tempel 1.

The spectra of the Comet Tempel 1 showed an increase in the ratio of

organics to water after the impact. The organic feature was also found to

be broader and contained more structures after the impact. This indicates

the presence of many new species that were below the detectable limit prior

to the impact.

The 3.4 µm cometary feature has also been seen from interstellar clouds,

molecular clouds, HII regions and so on. But the characteristic feature

is different indicating the variation of carbonaceous material present in

different environments.

9.5.4. Ice signature

All the available evidence indicates that H2O is the major component

of the nucleus of a comet. Therefore, it is interesting to look for ice grains

in the cometary comae.



Large size particles of ice grains could be stable against sublimation

around 1 AU. However, even with a small amount of absorbing material

present, the high temperature attained by the grain can entirely sublimate

the icy grain. Therefore it should be looked for in comets at large helio-

centric distances where the temperature of the grain is low enough that the

icy grains are present. The strong OH production seen in Comet Bowell at

r > 4 AU has been attributed to icy grains.

A more direct method of detecting the presence of icy grains in comets

is to look for the absorption bands of ice in the infrared region. Labora-

tory investigations have shown that ice band has features near 1.5,2.2 and

3.1 µm. Among these, 3.1 µm feature is the strongest. These bands have

been looked for in several comets. The observations on Comets Bowell and

Cernis at r ∼ 4 AU gave an indication of the presence of 3.1 µm absorp-

tion feature. A broad and shallow absorption feature appears to have been

detected in Comet Hale-Bopp when the comet was at 7 AU.

The strong water-ice band at 3.1 µm has been definitely detected for

the first time from Comet Tempel 1 from the infrared observations of the

ejecta from the Deep Impact experiment. This is shown in Fig. 9.19. This

indicate that large amount of icy grains from the nucleus of Comet Tempel

1 was lifted gradually after impact without being evaporated. This shows

that the excavated pristine material from a depth of 10 to 20 meters consist

of water-ice particles of sizes ∼ 1 µm.

9.6. Properties Derived from Direct Measurements

Comet Halley

The important information about the chemical composition of dust par-

ticles in Comet Halley has been obtained from the dust impact mass ana-

lyzer PUMA 1 and 2 on Vega and PIA on Giotto spacecrafts. Dust particles

striking a silver target placed in front of the mass spectrometer generate a

cloud of ions and the positive charge are mass analyzed, which indicates its

chemical composition. Several thousand mass spectra of dust particles were

recorded by the instruments. These studies indicated broadly three classes

of particles (Fig. 9.20): (1) mostly made up of light elements such as H, C,

N and O indicative of organic composition of grain called ‘CHON’ particles

(2) similar to CI Chondrites but enriched in carbon and (3) primarily O,

Mg, Si and Fe suggestive of silicate grains. Therefore, the Comet Halley



grains were found to be essentially composed of two end member particle

types - a silicate and a refractory organic material (CHON) in accordance

with the infrared observations.

Fig. 9.19 Comparison of the observed broad feature in the region of 3 µm from theejecta of Comet Tempel 1 with the model spectra of 0.5 µm diameter water ice particles.

There is a good fit (Sunshine, J.M. et al. 2007. Icarus, 190, 284).

The possibility of the presence of core-mantle structure of dust parti-

cles comes from the fact that on the average CHON ions appear to have a

higher initial energy than the silicate ions. The observed ion abundances

have been transformed to atoms based on laboratory calibration from the

knowledge of the ion yields with different projectiles. The resulting compo-

sition of Comet Halley’s dust indicate that the abundances of rock forming

elements in Halley’s dust are within a factor of two relative to the solar

system abundances. The abundances of H, C and O are more than that

of CI-chondrite believed to be the unaltered meteorites from the early so-

lar system. This can be interpreted to mean that the Halley dust is more

primitive than CI-chondrites. The puzzle of carbon depletion in comets also

seems to have been resolved as they are tied up in the refractory organics.

Due to uncertainties in converting the ion intensities to atomic abundances,

there could be a factor of two uncertainties in the final derived abundances.

The derived gas to dust ratio is ∼ 1.1 to 1.7. The estimated density of

silicate dominated grain indicates a value ∼ 2.5 gm/cm3, while CHON

dominated grain indicates a value ∼ 1 gm/cm3. Several refractory grains

show 12C/13C≈ 5000 which is drastically different from the normal value

∼ 89. This indicate that the carbon coming out of different nucleosynthesis

sites has been incorporated in the Comet Halley dust particles.



Fig. 9.20 Composition of dust particles seen by dust impact mass analyzer by Vega

spacecraft from Comet Halley (Kissel, J. et al. 1986. Nature, 321, 336).

The possible presence of polymerized formaldehyde (H2CO)n (also

called polyoxymethylene, POM) in the dust grains of Comet Halley has

been proposed based on the results of the positive ion cluster analyzer

(PICCA) experiment conducted aboard the Giotto spacecraft (Fig. 6.16).

The dust grains contain a large number of organic compounds (Table 13.8),

e.g., unsaturated hydrocarbons like pentyne, hexyne, etc; Nitrogen deriva-

tives like hydrocyanic acid etc; Aldehydes and acids like formaldehyde,

formic and acetic acid etc. Many more molecules are expected to be present

in the dust. It is quite possible that a large number of molecules might have

also been destroyed due to the high velocity of impact ∼ 78 km/sec.

Comet Wild 2

The stardust spacecraft flew past the nucleus of Comet Wild 2 within



a distance of 234 km on 2 Jan 2004. The particles ejected from the nucleus

of Comet Wild 2 were collected when they hit silica aerogel with a speed

of 6.12 km/sec. Stardust collected more than 1000 dust particles of sizes

in the range of about 5–300 µm. Stardust mission was meant for collecting

dust samples from Comet Wild 2 and bring back to earth for laboratory

investigation. The stardust spacecraft also contained a time-of-flight spec-

trometer called CIDA (cometary and interstellar dust analysis) which was

used for the study of the composition of dust particles in the coma. These

two studies have given detailed information about the composition of dust

particles of Comet Wild 2.

The CIDA instrument was similar to that in PUMA 1 and 2 on the

Vega spacecraft and PIA instrument on Giotto of Comet Halley. The CIDA

instrument is a time-of-flight mass spectrometer and is based on the fact

that the dust particles with enough velocity striking a silver target produce

either positive ions or negative ions depending on the voltage applied. The

flight time of these ions from the time of impact to the time of detection

give a direct measure of the mass of these ions.

The dominant ion detected from CIDA instrument was CN−. This

indicates the dominance of nitrogen-rich chemistry. It was found that O−

and OH− ions were minor in abundance which show that oxygen chemistry

play a minor role in these dust particles. Hence water-ice cannot be a major

constituent of these dust particles.

The importance of sulphur chemistry in the dust particles came out of

the detection of the two isotopic lines of SH− (mass-to-charge ratio m/z =

33 and 35). it is possible that sulphur atom which is generally attributed to

mineral phases of cometary dust could have come out of minerals such as

troilite to organic phase by some radiative process during the time (several

billion years) the dust particles spent in interstellar space and later in the

nuclei of comets.

In the positive ion mass spectra, three types of species have been de-

tected. They are CH+ ion(m/z = 13+), traces of N+, NH+,O+, OH+ and

unsaturated organic species containing some nitrogen.

Therefore studies based on CIDA instrument of Comet Wild 2 indicate

that H2O and CO were absent in the dust particles, nitrogen chemistry

appears to be dominant and sulphur must have come out of mineral phase

of the cometary dust.

The stardust samples collected from Comet Wild 2 have been subjected

to multiple experimental techniques. They show the presence of organic

aromatic matter(PAHs). As shown in Fig. 9.21 it contains benzene(78 amu,



C6H6), phenol(94 amu, C6H5OH), naphthalene(128 amu, C10H8) and their

alkylated derivatives, acenaphthylene(152 amu, C12H8), phenanthrene(178

amu, C14H10), pyrene(202 amu,C16H10), perylene(252 amu,C20H12) etc.

The features present at 101, 112, 155 and 167 amu could arise from O

and N substituted aromatic species. The higher mass spectra extending to

about 800 amu could be attributed to polymerization of smaller aromatics

as a result of chemical processing due to radiation processes, such as cosmic

rays, solar heating etc.

Glycine which is an amino acid and is an important constituent of living

systems has been detected for the first time in the dust particles of Comet

Wild 2.

Infrared spectra of dust particles show the presence of non-aromatic

functional groups as can be seen from the absorption features present

at 3322 cm−1(-OH), 3065 cm−1 (aromatic CH), 2968 cm−1(-CH3), 2923

cm−1(-CH2 -), 2855 cm−1(-CH3 and -CH2-) and 1706 cm−1(C=O).

The observed spectra indicate the presence of aromatic, aliphatic, car-

boxylic and N-containing functional groups. The organics in Comet Wild

2 is found to be considerably richer in O and N relative to both meteorite

organic matter and the average composition of Comet Halley.

Isotopic measurements have given some interesting results. D/H ratio

in some of the dust particles was found to be about three times larger than

the terrestrial value. D and H were heterogeneously distributed within

the samples and is associated with C. This indicates that it is organic in

character. The enrichment of 15N was also seen in some dust particles. The

higher value of D and 15N suggest that cometary material contain materials

from those of interstellar and/or protostellar environments.

All the studies carried out on dust particles from Comet Wild 2 has

shown the rich diversity of organic materials. They show a wide variety of

forms in molecular structure and molecular complexity. The distribution

of organics is heterogeneous inside as well as between the dust particles

indicating that they are unequilibrated material that experienced very little

processing after incorporating into dust particles. The enrichments in D and15N in the dust particles indicate the presence of presolar material.

9.7. Radiation Pressure Effects

The dust particles released from the nucleus are subjected to radiation

pressure forces which push them to different distances from the nucleus and



Fig. 9.21 Mass spectra of stardust sample C2115, Track 22 of Comet Wild 2. Thecomplex distribution of aromatic species covering molecular weights upto 500 can be

seen in (B). The expanded portion of mass range is shown in (C) and (D). The complex

nature of organics present can be seen (Clemett, S.J. et al. 2007. Lunar and PlanetaryScience, 38, 2091.

this ultimately gives rise to the observed dust tail as discussed in Chap. 7.

The dust tail is composed of various Syndynes and each curve is defined

in terms of a parameter β ≡ (1 − µ), which is the ratio of solar radiation

pressure to gravity. These Syndynes when projected on to the photographs

of the tail give a range of β values which encompass the observed tail. The

maximum value of β fixes the minimum size of the particles that can exist in

the tail. Others will essentially be pushed away from the system. From such



comparisons, it is possible to get the maximum value of β denoted as βmax

for various cometary tails. Table 9.6 gives a list of βmax values obtained

for various comets. They show that there are no particles in the tail which

Table 9.6 Values of βmax obtained for variouscomets∗ (r = 1 AU).

Comet βmax Comet βmax

1957 III 0.55 1970 II 1.9

1957 V 2 1970 II 3.8a

1962 III 2.5 1973 XII 0.8b

1965 VIII 1.1-1.4 1976 VI 2

1965 VIII 0.8 (farthest 1976 VI 2.5part) - -

1965 VIII 2.5 (near the Several 2.5

head) Comets

a. from IR photometry; b. fromColorimetry(∗Saito, K., Isobe, S., Nishioka, K.

and Ishii, T. 1981. Icarus 47, 351.)

are subjected to radiation pressure forces beyond about βmax ' 2.5. This

apparent cut-off in the value of β can be used to infer the properties of

the dust particles. However, the interpretation is not straightforward as it

involves various parameters of the grains, which means one has to invoke

various grain models. As an example, Fig. 9.22 shows a plot of the variation

of β as a function of the radius of the particle for different types of particles.

It can be seen that for small size particles most of the curves are flat except

for the silicate type (Basalt) of materials. The observed maximum value of

βmax ' 2.5 seems to lie in between these two general shapes of the curves.

9.8. Summary

We may now briefly summarize the main characteristic properties of

cometary grains based on the discussion presented so far. In order to re-

duce the ambiguity, it is important as well as necessary to consider various

types of observations which have to be satisfied simultaneously by any grain

model. However, it may not be practicable in all the cases. In the absence

of such detailed studies, it is even worthwhile to restrict the parameters



Fig. 9.22 The variation of radiation pressure force (1 − µ) ≡ β with the radius of

the particle for three types of material is shown. (Adapted from Saito, K., Isobe, S.,Nishioka, K. and Ishii, T. 1981. Icarus, 47, 351.)

which define the grain characteristics within certain possible ranges.

The optical continuum measurements, the polarization measurements

as well as scattering phase function in the visual region give a limit for the

refractive index as 1.3 < n < 2.0 and k ≤ 0.1.

These are based on single particle models or multi-component models

based on different compositions and size distribution functions. Models

based on aggregaates of large number of monomers with sizes of ∼ 0.1 µm

composed of several compositions have also been considered. Any realis-

tic model has to explain all the observations pertaining to cometary dust.

Attempts are made to develop such a model.

The silicate nature of grain came from the detection of 10 and 20 µm fea-

tures from comets. The evidence for the second component to the cometary

grain namely organics, came from the discovery of CHON particles in

Comet Halley as well as the detection of 3.4 µm feature in the infrared



observation. Therefore, the two major components of the cometary dust

are silicates and some form of carbon.

The true mineralogical composition of cometary dust particles came

from the mid and far-infrared observations of Comets Halley, Hale-Bopp,

Tempel 1 and Wild 2 and the laboratory studies of cometary dust of Comet

Wild 2. They showed the presence of silicates of various kinds including

crystalline type, carbonates, water ice, amorphous carbon, sulphides, PAHs

etc. Calcium-Aluminium-rich inclusions, the oldest samples in the solar sys-

tem containing minerals in abundance, which condensed around 1400 K was

seen. The dust particles also show rich diversity of organics. The detection

of Glycine in cometary dust indicate the presence of ingredient for the for-

mation of living systems. There is enrichment of 15N and D in the dust par-

ticles indicating the presence of presolar grains. Therefore cometary dust is

an unequilibrated heterogeneous mixture of materials containing both high

and low temperature condensates and also contains presolar grains. This

also indicated for the first time, that large scale mixing must have taken

place in the solar nebula.

Problems

1. Derive an expression for the variation of the temperature of a black

body as a function of distance from the Sun. Assume that the energy

distribution of the Sun can be represented by a black body of tem-

perature 6000K. What will happen if the body is a planet with no

atmosphere or if there is an atmosphere?

2. Calculate the grain temperature for graphite grains of a = 0.2 µm at

1 AU from the Sun. How much does this value differ from that of black

body temperature at the same distance?

3. Suppose in the above problem, the grain has some impurity which gives

Qabs = 1 at 1 mm. What will happen to the temperature of the grain?

4. Calculate the number of electrons and atoms or molecules required to

explain the observed scattered radiation in comets at r = 1 AU and for

λ = 5000 A. Take the value of the scattered intensity to be 50% of the

solar intensity at that wavelength. What is the expected wavelength

dependence of these?

5. Suppose the heavy elements like O, C, N hit the grain and stick to it

with a probability α. If the temperature of the gas is T and assuming

the mean velocity to be given by the Maxwellian distribution, deduce



the expression for the rate of growth of the grain. What is the time

required for an ice grain of initial radius of 0.01 µm to grow to 1 µm,

if α = 1, T = 100K and nH = 10/cm3

6. A grain releases a molecule and reduces in size due to protons hitting

the surface of the grain. Deduce an expression for the rate of decrease

of grain size with time, if β is the probability for the release of a lattice

molecule when the proton hits the surface. What is the time required

to destroy completely an ice grain of size 1 µm with NH = ne = 10/cm3,

T = 104 K and β = 0.1?

7. Assuming solar constant of 1.4 × 106ergs/cm2

sec and spherical com-

pletely absorbing particles of density 3.0, calculate the limiting radii of

particles to be retained in the solar system released from Comet Encke

at aphelion distance of 4.10 AU and perihelion distance of 0.34 AU.

8. Compare the importance of radiation pressure on the motion of the dust

compared with the force of gravity at the orbit of Venus. Calculate the

time required for the dust to be driven out to the distance of the Earth

by radiation pressure alone.

References

The following paper gives a good account on dust

1. Kolokolova, L, Hanner, M.S., Levasseur-Regourd, A.-Ch et al. 2005.

In Comets II, eds., M.S. Festou, H.U. Keller and H.A. Weaver, Univ.

Arizona Press, Tucson, P.577.

The early work on continuum studies can be found in these two references:

2. Liller, W. 1960. Ap. J. 132 867.

3. Remy-Battiau, L. 1964. Acad. r. Belg. Bull. el. Sci. 5eme Ser. 50

74.

The following papers may be referred for later work:

4. Jewitt, D. C. and Meech, K. J. 1986. Ap. J. 310, 937

5. Hanner, M.S. 2003. JQSRT, 79-80, 695.

6. Wolf, M.J., Clayton, G.C. and Gibson, S.J. 1998. Ap.J., 503, 815.

7. Mann, I., Kimura, H. and Kolokolova, L. 2004. JQSRT, 89, 291.

8. Mason, C.G., Gehrz, R.D., Jones, T.J. et al. 2001. Ap.J., 549, 635.

The determination of grain albedo can be found in the following papers

9. O’Dell, C.R. 1971. Ap. J., 166, 675.

10. Hanner, M.S. and Newburn, R.L. 1989. Astron, J., 97, 254.

A(θ)fρ was introduced in the following paper



11. A’Hearn, M.F., Schleicher, D.G., Feldman, P.D., et al. 1984. Ap.J.,

89, 579.

The early measurement of polarization was done by

12. Bappu, M.K.V. and Sinvhal, S.D. 1960. M N 120 152.

The existence of negative polarization in comets was shown in the following

two papers:

13. Kiselev, N.N. and Chernova, G.P. 1978. Soviet Astron. 22 607.

14. Weinberg, J.L. and Beeson. D.E. 1976. Astron. Astrophys. 48 151.

For later work the following may be consulted

15. Dollfus, A. 1989. Astr. Ap. 213, 469.

16. Mukai, T., Mukai, S. and Kikuchi, S. 1987. Astr. Ap. 187, 650.

17. Harrington, D.M., Meech, K. Kolokolova, L., et al. 2007. Icarus, 187,

177.

The possible division of comets based on observed linear polarization is in

the following papers

18. Levasseur-Regourd, A. -Ch., Hadanicik E. and Rehard, J.B. 1996. As-

tron. Astrophys., 313, 327.

19. Levasseur-Regourd, A. -Ch., Zolensky, M. and Lasue, J. 2008. Plane-

tary and Space Science, 56, 1719.

20. Jockers, K., Kiselev, N., Bonev, T. et al. 2005. Astron. Astrophys.,

441, 773.

Circular polarization could be due to organics is discussed in the following

paper

21. Rosenbush, V. Kolokolova, A., Lozarian, A. et al. 2007. Icarus, 186,

317.

In situ studies are discussed in the following papers

22. Mazets, E.P., Aptekar R.L., Golenetskii, S.V. et al. 1986. Nature, 321,

276.

23. McDonnell, J.A.M., Lamy, P.L. and Pankiewicz, G.S. 1991. In Comets

in the Post-Halley Era, eds. R.L. Newburn, M.Neugebauer and J. Rahe,

Kluwer Academic Publishers, P. 1043.

24. Nordholt, J.E., Reisenfeld, D.B., Wiens, R.C. et al. 2003. Geophys.

Res. Lett., 30, 18.

25. Tuzzolino, A.J., Economou, T.E., Clark, B.C. et al. 2004. Science,

304, 1776.

The first infrared measurements made on Comet Ikeya-Seki is reported by

26. Becklin, E E and Westphal, J A. 1966. Ap. J., 145 445.

The following paper reports the first detection of 10 µm emission feature

in Comet Bennett.



27. Maas, R., Ney, E.P. and Woolf, N.J. 1970. Ap. J.,160, L101.

Interpretation of infrared observations and mineralogy of grains can be

found in the following papers

28. Krishna Swamy, K.S. and Donn, B 1968. Ap J 153, 291.

29. Krishna Swamy, K.S., Sandford, S., Allamandola, L.J. et al. 1988.

Icarus, 75, 351.

30. Oishi, M., Okuda, H. and Wickramasinghe, N C 1978. Publ. Astro.

Soc. Japan, 30, 161.

31. Crovisier, J., Brooke, T.Y., Leech, K. et al. 2000. Thermal Emission

Spectroscopy and Analysis of Dust Disks and Regoliths, eds. M.L. Sitko,

A.L. sprague and D.K. Lynch, ASP Conference Ser. Vol. 196, P.109.

32. Wooden, D.H., Harker, D.C., Woodward, C.E., et al. 1999. Ap.J.,

517, 1034.

Study of complex mineralogy of dust is in the following papers

33. Crovisier, J., Leech, K., Bockelee-Morvan, D. et al. 1997. Science, 275,

1904.

34. Lisse, C.M., Kraemer, K.E., Nuth III, J.A. et al. 2007. Icarus, 187,

69.

35. Hanner, M.S. and Bradley, J.P. 2005. In Comets II, eds. M.C. Festou,

H.U. Keller and H.A. Weaver, Univ. Arizona Press, Tucson, p. 555.

The following papers discusses the complex nature of organics

36. Sandford, S.A. 2008. In Organic Matter in Space, Eds. S. Kwok and

S.A. Sandford, Cambridge University Press, Cambridge, P.299.

37. Brownlee, D., Tsou, P., Aleon, J. et al. 2006. Science, 314, 1711.

38. Kissel, J. Krueger, F.R., Silen, J. et al. 2004. Science, 304, 1774.

39. Kissel, J. and Krueger, F.R. 1987. Nature, 326, No. 6115, 755.

40. Disanti, M.A., Villanueva, G.L., Bonev, B.P. et al. 2007. Icarus, 187,

240.

41. Biver, N., Bockelee-Morvan, D., Boissier, J., et al. 2007. Icarus, 187,

253.

42. Clemett, S.J., Nakamura-Messenger, K., Mckay, D.S. et al. 2007. Lu-

nar and Planetary Science, 38, 2091.

The first detection of Glycine in Comet Wild 2 is reported in the following

paper

43. Elsila, J.E., Glavin, D.P. and Dworkin, J.P. 2009. Meteorites and Plan-

etary Science, 44, 1323.

The observed C-H(3.4 µm) feature in Comet Tempel 1

44. A’Hearn, M.F. 2005. In Asteroids, Comets and Meteors, eds.

D.Lazzaro, S. Ferraz-Mello and J.A. Fernandez, Cambridge Univ.



Press, p. 33.

The first definite detection of strong 3.1 µm ice dust feature in Comet Tem-

pel 1 is reported in the following paper

45. Sunshine, J.M., Groussin, O., Schultz, P.H. et al. 2007. Icarus, 190,

284.


CHAPTER 10

Ion Tails

The ion tails of comets provide a unique and natural place for the study

of various plasma processes. Many of the features seen in the ion tails

of comets change on a short-time scale as well as on a long-time scale.

The formation of ion tails is basically due to the interaction of the solar

wind plasma with the cometary plasma. The detailed dynamical model

calculations for such an interaction process had indicated the presence of

Large Scale Structures like bow shock, ionosphere and so on. The in situ

measurements of Comets Giacobini-Zinner, Halley Borrelly, etc. showed

not only the presence of these features but also indicated the plasma to be

highly complex, containing instabilities, waves, turbulence and so on. Some

of these aspects will be discussed here. In addition, some of the large scale

features that have been seen in the ion tail of comets will also be discussed.

10.1. Evidence for the Solar Wind

The existence of the solar wind was first postulated based on the study

of the ion tail of comets. The model was put on a firm footing with the

work of Parker based on the hydrodynamic expansion model of the solar

corona. The model predicted in detail the expected nature of the solar wind

as well as the resulting shape of the interplanetary magnetic field. These

were confirmed later on with observations made through satellites.

The idea of the solar wind came from the photographic observations of

comets which showed that certain features like knots in the ion tails were

moving away along the tail. The velocity and the acceleration of the features

obtained from successive photographs showed clearly that these features

were moving in the anti-solar direction with high velocities. The typical

velocity is about 100 km/sec. A measure of the deceleration is generally

303



expressed in terms of the effective gravity parameter (1 − µ) as defined in

Chap. 7. The typical values of (1−µ) ≈ 102 were quite commonly observed

in the ion tails of many comets. This indicates an outward force which is

102 times larger than the solar gravity. One force which could operate and

give rise to this motion on an atom or a molecule is the radiation pressure.

The expression for the radiation pressure is given by

Frad =

(Fc

)(πe2

mcf

)(10.1)

where F is the solar flux at the comet and all other quantities have their

usual meanings. It was found that the Eq. (10.1) fails to explain the

observed accelerations of features in the ion tail by several orders of mag-

nitude. This led Biermann to hypothesize the existence of the solar wind,

i.e., the high velocity corpuscular radiation coming from the Sun, which

accelerates the ions in the tail through the momentum transfer. In fact he

developed a very simple expression for this interaction and obtained the

equation

dvidt≈ e2Neveme

σmi. (10.2)

Here the quantities with the subscript e refer to electrons and i to ions, Neis the electron density and ve is the velocity of electrons, σ is the electrical

conductivity. In order to explain the observed accelerations, the above

equation requires the solar wind velocity to be of the order of a few hundred

kilometers per second and the density of about 600 electrons/cm3. These

values may be compared with the present day velocity of about 400 to

500 km/sec and the density of about 5/cm3.

10.2. Dynamical Aberration

More support for the solar wind hypothesis came from the study of the

orientation of ion tails of comets. The tail axes of ion tails are found to lag

behind by a few degrees with respect to the radius vector. The lag arises

due to the resultant effect of the solar wind velocity and the velocity of the

comet. This is known as the dynamical aberration. Figure 10.1 shows the

geometry of the comet tail on the plane of the sky. Here r, t and −v denote

the radius vector, the tail vector and the negative velocity vector of the

comet. For the interpretation of ion tail orientation, it is convenient to use

the plane of the comet’s orbit as the reference frame. The observational


Ion Tails 305

Fig. 10.1 The geometry of the comet tail (dashed lines) projected on to the plane of thesky. The position angles φ, θ and ψ denote the prolonged radius vector, the comet tail

and the direction of comet’s velocity back along the orbit respectively. (Belton, M.J.S.

and Brandt, J.C. 1966. Ap. J. Suppl. 13, 125).

data is the position angle θ of the tail axis on the plane of the sky. From a

knowledge of the various quantities as shown in Fig. 10.1, the orientation

of the tail in the comet’s orbital plane can be calculated (Fig. 10.2) with

the assumption that it also lies in the same plane. The aberration angle ε

is then the angle of the tail with respect to the radius vector. The basic

equation used for the interpretation of dynamical aberration observation is

given by

t = w − v (10.3)

where t is the tail vector which lies in the orbital plane of the comet, w is

the solar wind velocity and v is the comet’s orbital velocity. Physically the

above equation means that the direction of the tail is the direction of the



Fig. 10.2 The geometry of the comet tail projected on to the plane of the comet’s orbit.The aberration angle ε and the angle γ are shown. (Belton, M.J.S. and Brandt. J.C.

1966. op. cit.).

solar wind as seen by an observer riding on the comet. If the solar wind

velocity is resolved into radial (wr) and azimuthal (wφ) components, the

aberration angle for a comet near the solar equator is given by

tan ε =v sin γ − wφ cos i

wr − v cos γ(10.4)

where i is the inclination of the comet’s orbit to the plane of the solar

equator and γ is the angle between the radius vector and the direction of


Ion Tails 307

v. All the quantities in the above equation are known except for wr and

wφ. Since the value of wφ is small, the value of wr can easily be calculated

from Eq. (10.4). Since the observations on comets cover a wide variety of

situations, it is possible in principle to get not only wr and wφ but also

their variations. If wφ has an appreciable value, it should show up in the

aberration angle between the direct (D) and retrograde (R) comets. The

above method has been applied to about 60 comets covering approximately

1600 observations. The resulting averages for the aberration angles for

direct and retrograde comets are

〈ε〉D = 3.7

and

〈ε〉R = 5.5.

Assuming wr and wφ are the same for the two cases and using the average

values for other parameters, the resulting mean values for wr and wφ are

〈wr〉 = 450± 11 km/sec

and

〈wφ〉 = 8.4± 1.3 km/sec.

In the above discussion, it was assumed that the tail of the comet lies

in the orbital plane of the comet. This assumption can be relaxed and

the velocity field of the solar wind can also be taken into account. The

following variations for wφ and wθ which are found to be consistent with

the observational and theoretical evidences have been used in the comet

tail orientation problem

wr = constant, wφ = wφ,o(cos b)2.315

r(10.5)

and

Wθ = wm sin 2b (10.6)

where b is the solar latitude, wφ,o and wm denote the azimuthal speed

in the plane of the solar equator at 1AU and the maximum value of wθ.

The results of analysis are given in Table 10.1. It is also possible to get

the minimum solar wind speed which actually refers to the larger values

of ε. The minimum value obtained for the solar wind speed is around

225 ± 50 km/sec. All these results are in good agreement with the direct

space probe measurements. The presence of the azimuthal component of the



Table 10.1 Parameters of solar wind derived from ion tails.

Parameter Published sample Total sample (includes unpublished data)

ωr (km/sec) 402 ± 12 400 ± 11ωφ,o(km/sec) 7.0±1.8 6.7±1.7ωm(km/sec) 2.6±1.2 2.3± 1.1

Number of

observations 678 809

(Brandt, J.C. and Mendis, D.A. 1979. In Solar System Plasma Physics, Vol. II,

eds. C.F. Kennel, L.J. Lanzerotti and E.N. Parker. Amsterdam: North Holland

Publishing Company.)

solar wind was also first deduced based on the study of ion tails which was

later confirmed through the direct space probe measurements, vindicating

the simple aberration picture of the ion tails of comets. It is also remarkable

that it can give so much of information without a detailed knowledge of the

interaction between the solar wind and the tail plasma.

From the above discussion, it is apparent that the derived results based

on the use of Eq. (10.3) are in good agreement with the gross observed prop-

erties of plasma tail of comets. Having established the validity of Eq. (10.3),

it is now possible to reverse the problem and calculate approximately the

expected general shape of the plasma tail from the same equation. This

formalism is known as the Wind Sock Theory. Therefore the existence of

solar wind was well established from the cometary studies before its detec-

tion by spacecraft at a much later time. The above studies indicate the

importance of knowing the nature, structure and composition of solar wind

for a full understanding of the comet and solar wind interaction.

10.3. Theoretical Considerations

The comet acts as an obstacle to the free flow of the solar wind. There-

fore one can approach the problem of the interaction of the solar wind with

the cometary atmosphere purely from the theoretical considerations. The

presence of fine structures in the plasma tail of comets indicates the exis-

tence of magnetic field in comets for its confinement. Otherwise the fine

structure will be washed out due to thermal motions.

Therefore the theoretical study of the interaction between the solar wind

and cometary ions is a complicated problem as the interaction is coupled


Ion Tails 309

through the interplanetary magnetic fields. The basic idea of the comet-

solar wind interacation was provided by Alfven in 1957 and is shown in

Fig. 10.3. When the solar wind with its frozen-in-magnetic field encounters

Fig. 10.3 A schematic representation of the “piling up” and “curling up” of interplan-etary magnetic field by the coma of the comet. The gradual formation of the ion tail is

shown in (a), (b), (c) and (d). (Adapted from Alfven, H. 1957. Tellus 9, 92).

a comet containing ionized molecules, these molecules cannot cross the

field lines. Instead they spiral around the magnetic lines and hence they

are attached to the field lines. The cometary ions trapped in the field line is

called ‘mass loading’. Since the cometary ions have low velocities, they are

essentially at rest with respect to the solar wind speed of ∼450 km/s. This

reduces the flow speed. This effect is stronger near the nucleus than well

away from the comet. These field lines wrap around the comet’s ionosphere

and it is finally dragged into the tail as shown in Fig. 10.3. It also helps



in transfer of momentum betwen solar wind plasma and the cometary ions

and this formalism can explain in a natural way the presence of narrow

and straight streamers in the ion tail of comets. This is the basis of all the

plasma models.

In a simple model, the plasma can be assumed to act as a fluid in the

sense of fluid dynamics, allowing the use of simple hydrodynamic methods

to predict the flow pattern of the plasma. Even without detailed modelling,

it was suggested around 1964 that the expansion of the ionized coma gas

should act as an obstacle to the supersonic solar wind flow which should

result in a bow shock at a distance of around 104 to 105 km from the nu-

cleus. The particle distribution functions can usually be determined from

the conservation laws, namely, continuity, momentum and energy equations

including the effect of electric and magnetic fields. These are to be sup-

plemented with the equations for the magnetic field and the electric field.

These equations constitute the magnetohydrodynamic equations (MHD) for

the plasma and provide a good description of most of the plasma processes

in the coma. The results of accurate one-and-two-dimensional models with

spherical gas flow showed that a weak bow shock is expected with a Mach

number 2. In recent years, more realistic and sophisticated models have

been constructed, which takes into account various physical processes in-

cluding chemistry in the neutral and ionized coma and its interaction with

the solar wind. The interaction of the ionized coma with the solar wind

lead to deviations from spherical symmetry. The assumption of axial sym-

metry is also destroyed due to the embedded magnetic field. Therefore a

detailed description of the region would require a 3-dimensional magneto-

hydrodynamic model. These MHD models describe the macroscopic flow

patterns and the characteristic boundaries expected from such interactions.

The result of one such detailed 3-dimensional MHD calculation is shown in

Fig. 10.4. The model takes into account momentum exchange from ion-

neutral collisions, photoionization, mass-loading through ion pick up and

the Lorentz forces of the transverse magnetic field. The simulation also

takes into account the detailed physics and chemistry of the coma together,

as they are intimately connected.

The results show that a bow shock occurs around 5× 105 km from the

nucleus for Comet Halley parameters. The ions are heated by the shock

to around 2.2× 106 K. However they cool down very fast due to collisions

and charge exchange. The temperature of electrons is not affected by the

shock and they lose their energy due to electron impact reaction. A sharp

boundary exists at a distance of around 4800 km where the flow velocity


Ion Tails 311

Fig. 10.4 Variation of physical quantities as a function of distance from the nucleus (R

in km) for axisymmetric model (dashed lines) and the three dimensional model (solidlines). Ti=ion temperature (K), Te=electron temperature (K), v=velocity (km/sec),

N=Number density (/cm3), B=magnetic field strength (nT) and µ=mean molecular

weight (Wegmann, R., Schmidt, H.U., Huebner, W.F. and Boice, D.C. 1987. Astr. Ap.187, 339).

falls steeply. This is the contact surface which separates the contaminated

solar wind from the pure cometary plasma inside. Therefore at this surface

two plasmas of different origin converge. The magnetic field is compressed

by the shock and it reaches a value of about 53 nT in the stagnation zone.

The value drops suddenly at the contact surface. The variation of physical

parameters as shown in Fig. 10.4 arise as a result of balance of different

effects.

The overall morphology of the flow of plasma near a comet is schemat-

ically shown in Fig. 10.5. They can be understood as follows. At large

distances from the comet the interaction of the solar wind is collisionless



Fig. 10.5 Schematic representation of the global morphology produced by the interac-tion of the solar wind with the cometary atmosphere shows the flow pattern and the

presence of various discontinuities arising from such an interaction (Mendis, D.A. 1988.

Ann. Rev. Astr. Ap. 26, 11).

as the cometary material is neutral. Once the ions are created, they are

accelerated due to the interplanetary magnetic field (B) and the motional

electric field of the solar wind, E=-Vsw× B where Vsw is the solar wind

velocity. The most important source of ionization of the gas is the pho-

toionization by the extreme ultraviolet radiation. The charge exchange

transfer of solar wind protons with cometary neutral also contribute to the

ionization process. The ions formed move with a typical speed of around

1 km/sec in the cometary frame of reference, while the solar wind velocity

is 400 km/sec. The newly formed ions which are initially at rest are accel-

erated by the motional electric field of the solar wind leading to a cycloidal

motion with both gyration and E×B drift motion. The gyration speed

depends upon the angle between the solar wind velocity and the magnetic

field. This assimilation of the cometary ions into the magnetized solar wind

is called ‘mass loading’ of the inflowing solar wind. The dynamics of the

solar wind containing cometary ions depend upon the overall pressure and

density associated with these ions. The thermal speed of ions, picked up by

the solar wind, is of the order of solar wind speed. The picked up energy of


Ion Tails 313

the ions in the upstream in the solar wind reference frame is typically about

10–20 keV for O+, whereas the solar wind protons have thermal energies

∼ 10 eV. Therefore the picked up ions are quite hot and the pressure due

to these ions dominate the total pressure even at large cometocentric dis-

tances. With the decrease in cometocentric distance, more and more ions

are added to the solar wind and hence the total pressure increases. As solar

wind progressively gets ‘mass loaded’, due to momentum conservation, the

solar wind slows down. As the solution of the fluid equation cannot go

continuously from supersonic to subsonic state, this results in a weak bow

shock at a critical level of mass-loading. This depends crucially on the mean

molecular weight. Therefore the cometary bow shock arises purely as a re-

sult of mass-loading process and is unique among the solar system objects.

Many of the energetic ions like O+, C+ and H+ are generated upstream

of the comet. Moving inwards from the shock, the solar wind continues to

interact with the cometary material of increasing neutral density. These

outflowing neutrals play an important role in slowing down the incoming

solar wind through collisions. Therefore cometary ions which are produced

down-stream of the shock are less energetic (E ∼ 1 keV) compared to those

produced upstream of the shock. In a real situation there is a slow gra-

dation in the energy of the picked up ions and hence in the solar wind

reference frame, energy of the pick-up ions decreases with decreasing come-

tocentric distance. Therefore the plasma found in the inner regions of coma

contains ions of different populations made up of energetic ions picked up

far upstream, and relatively cold ions picked up locally Finally, the velocity

decreases rapidly, giving rise to a sharp boundary called ‘cometopause’,In

this transition region, collisions dominate. Near ‘cometopause’, the hot and

warm cometary ions disappear from the plasma through charge exchange

collisions with the neutrals. Inside the ‘cometopause’, the speed is very low,

the ions are abundant and are modified by collisions and chemistry. This

also leads to compression of the magnetic field giving rise to the magnetic

barrier. The boundary of the field free cavity is called ‘diamagnetic cavity

boundary surface’. It is also called contact surface, the ionopause or the tan-

gential discontinuity. This boundary separates the purely cometary plasma

and the mass-loaded solar wind plasma. The basic physical mechanism of

its formation is due to the balance between the outward ion-neutral fric-

tional force (due to the flow of neutrals past stagnated ions) and an inward

directed electromagnetic (J × B) force. Here the mass loading term is less

important. The presence of another shock inside the cometary ionopause

has been suggested in order to decelerate the supersonic outward flowing



cometary ions and divert them into the tail.

10.3.1. Comparison with observations

The in situ measurements of Comets Giacobini-Zinner, Halley and Bor-

relly which provided detailed data, confirmed the large scale picture of the

solar wind interaction with comets at close heliocentric distances. Out of

the three comets, extensive and detailed in situ data exists for Comet Halley

due to six spacecrafts passing close to the nucleus of this comet.

The ICE spacecraft passed by Comet Giacobini-Zinner on Sept.11, 1985.

The observations indicated the prersence of a bow shock around 130,000 km

from the nucleus. The detection of the high energy particles and plasma

waves was an important result that came out of these studies. These high

energy ions are basically the ions that were picked up by the solar wind

magnetic field and accelerated as they are dragged by the solar wind. This

process generates plasma waves. The curling of the magnetic field was

also seen in the observations. The reversal of the magnetic polarity was

also seen from spacecraft measurements. Far from the nucleus, plasma had

typical solar wind speed of about 500 km/s. The density was around 5

ions/cm3 and an electron temperature of around 250,000 K. Around the

closest distance of the spacecraft to the nucleus of 7800 km, the flow speed

was about 30 km/sec, electron density of 600/cm3 and electron temperature

of 15000 K. The measured ions were mostly water-group ions, such as H2O+

and H3O+.

While the bow shock of Comet Giacobini-Zinner was not well defined,

the presence of a bow shock at a distance ∼ 1 × 106 km from the nucleus

in Comet Halley was clearly identified by instruments aboard spacecrafts,

measuring plasma, magnetic field and plasma wave. The detection of bow

shock in Comet Halley has resolved the controversy about its existence. The

spacecraft measurements, has also given detailed information on the struc-

ture of the bow-shock. For example, the water group ions O+, OH+, H2O+

and H3O+ showed a sudden enhancement following the bow shock. Inside

the bow-shock, in the cometo-sheath region the flow speed was found to de-

crease continuously due to ion pick up and the plasma becomes more dom-

inated by cometary ions. The phase space density distribution of cometary

H+ ions showed a shell-like structure at a distance of around 8 × 106 km

upstream of the bow-shock in Comet Halley. This pattern arises due to the

fact that the velocity distribution of pick up ions which should be in the

form of a ring with velocity component v⊥ = Vsw sinα perpendicular to


Ion Tails 315

the interplanetary magnetic field and a drift velocity v11 = Vsw cosα along

B and relative to the solar wind is unstable and leads to various plasma

instabilities. As a result of wave-particle scattering, the H+ ions diffuse

along the sphere transforming the ring-like structure to shell-like structure.

Many of these features have also been seen from Comet Borrelly from

measurements carried out with Deep Space 1.

The existence of cometopause at a distance of around 105 km came

from the observation that the density of the solar wind protons was found

to decrease rapidly approaching this distance, while the comet ion density

increased rapidly with 1/R2 radial dependence for R < 105 km (Fig. 10.6).

At a cometodistance ∼ 1.4 × 105 km there was also a sudden decrease of

Fig. 10.6 Number density profiles of water group ions and solar wind protons as mea-

sured by Giotto ion mass spectrometer (see Ip, W.-H. 1989. Ap. J. 343, 946).

electrons with energies ≥ 10 eV while the magnetic field strength increases



from 6 nT to 26 nT. For radial distances < 2 × 105 km the dominant

ions change progressively from O+, OH+ and H2O+ to H3O+ (Fig. 10.7).

The dominant ions inside 104 km are H2O+ and H3O+ which continuously

increase towards the nucleus as indicated by the models. The ions C+ and

S+ were also found to be very abundant.

Fig. 10.7 The variation of abundances of several species as measured by ion mass spec-trometer on Giotto from Comet Halley. R is the distance of Giotto from the nucleus and

the time refers to UT at the ground station (Balsiger, H. et al. 1986. Nature 321, 330).

The Giotto measurements of magnetic field, plasma temperature and

plasma speed is compared with that of model calculations in Fig. 10.8 and

Fig. 10.9. There is overall agreement between the models and observations

The ion temperature dropped from 2600 K to 450 K at a distance of around

4700 km. These results confirmed the presence of tangential discontinuity.

The interesting result that Giotto magnetometer measurements showed is

that there was a dramatic drop in the magnetic field ∼ 50 nT to zero

at cometocentric distance ∼ 4600 km from the nucleus inbound. In the

outbound the magnetic field of zero rose to about 65 nT at distance of about


Ion Tails 317

Fig. 10.8 The Giotto measurements of variation of magnetic field is compared with

modeled results (Gombosi, J.I., De Zeeuw, D.L. and Haberte, R.M. 1996. J. Geophys.

Res., 101: 15, 233).

3800 km. The region bound by these two distances is devoid of magnetic

field. The upper limit for the magnetic field strength in the stagnent plasma

region can be estimated from balance between the magnetic pressure and

the solar wind ram pressure

i.e.,B2s

2µo= nswmswV

2sw (10.7)

Using for solar wind number density nsw ∼ 5/cm3 and Vsw ∼ 400 km/sec,

the calculated value of Bmax ∼ 60 nT. This is what is observed.

The width of the field-free cavity is about 8500 km. The boundary of

the cavity separate the two regions of the plasma wherein the inner side

containing mainly pure cometary plasma while the outer region contain-

ing a mixture of cometary plasma and solar wind plasma. In essence the

outflowing cometary plasma hinders the flow of solar wind with its mag-

netic field inside this surface. The location at which the inner edge of the

ionopause occurs can also be estimated from the balance between the mag-

netic force and the ion-neutral drag force. This gives a distance ∼ 4335 km

for inbound and ∼ 3470 km for the outbound for the Giotto encounter.

These are in reasonable accord with the observed values mentioned above.

The general results of plasma measurements carried out in Comet Bor-

relly are quite similar to those of Comet Halley. The approximate composi-

tion as determined from the closest approach data is as follows: 63% OH+,



Fig. 10.9 The Giotto measurements of variation of speed and temperature of ions are

compared with modeled results (Gombosi, J.I., De Zeeuw, D.L. and Haberte R.M. 1996.J. Geophys. Res., 101:15, 233).

25%H2O+, 2.5%C+, 2%N+ and 8%CH+3 .

10.4. Instabilities and Waves

It was suggested that there could be some instabilities present in the

plasma due to the mass-loading of the solar wind by the newly created ions.

The nature of the pick up cometary ions by the solar wind depend upon

the orientation of the solar wind flow to the interplanetary magnetic field

(IMF)B. When the IMF is orthogonal to the solar wind Vsw, i.e. B⊥ Vsw the


Ion Tails 319

Vsw×B motional electric field makes the newly created ions gyrate around

the magnetic field. In this case the anisotropy in their gyro velocities (as

they can have both the zero velocity and twice the solar wind velocity),

can give rise to three types of low frequency instabilities, namely the ion-

cyclotron instability, a parallel propagating nonoscillatory mode and a fluid

mirror instability. In the other extreme case, when the IMF is parallel to

the solar wind velocity vector i.e. B‖ Vsw, there is no Vsw × B solar wind

force acting on the cometary ions. For such a situation the ions form a beam

in the solar wind plasma frame moving at a velocity −V sw relative to the

ambient plasma. This can lead to two types of instabilities, a right handed

resonant helical beam instability and a nonresonant instability. However

the coupling between the solar wind and the cometary ions becomes com-

plicated if the solar wind flows obliquely to the magnetic field at an angle

θ. Therefore, in principle, various kinds of plasma instabilities and waves

could be generated due to the ion pick up by the solar wind, depending

upon the complex local conditions of the plasma. The in situ plasma and

magnetic field measurements of Comets Giacobini-Zinner and Halley by the

spacecrafts have shown the presence of a large number of waves of various

kinds. They are highly complex and difficult to interpret. Only a few of

these results will be mentioned here.

One of the main results that came out of the study of the Comet

Giacobini-Zinner by the ICE spacecraft is the detection of the water group

(16 to 19 amu) ion cyclotron frequency from spectral power analysis. This

has also been detected in Comet Halley observations. Several peaks with

cyclotron fundamental at 7 mHz and its harmonics at 14, 21, 29 and 39 mHz

has been seen in the cross-spectral densities of the solar wind proton velocity

and magnetic field. The fundamental mode at 7 mHz is linearly polarized

and the higher harmonics are either linearly or highly elliptically polarized.

The cometary influence extends to a distance of around 2× 106 km.

Waves have been detected upstream of the bow shock for Comets

Giacobini-Zinner and Halley. A variety of polarizations have been detected

from circularly polarized waves propagating at large angles relating to the

ambient field to highly elliptical polarized waves. The detection of short

duration magnetic pulses during θ = 90 intervals was rather an unex-

pected result. The field variations have a duration of around 6 to 7 sec and

is comparable to the proton cyclotron frequency in the spacecraft frame.

These pulses are mostly transverse oscillations.

At distances ∼ 3 × 104 to 18 × 104 km from the comet the magnetic

field experiment detected mirror mode waves which are characterized by



irregular dips in the magnetic field. Similar structures have been seen

from Giotto Spacecraft in Comet Halley and ICE on Comet Giacobini-

Zinner. Higher frequency whistlers have been seen near the bow shock of

Comet Giacobini-Zinner and it is an integral part of the magnetosonic wave.

Several mechanisms for the formation of whistlers have been proposed such

as through generation of dispersive whistlers derived from hybrid simulation

results, generation of pick up of heavy ions and protons at the distorted

steepened fronts of the magnetosonic waves and so on.

The waves of extremely low frequency (ELF) in the region 10 to 1500 Hz

and of very low frequency (VLF) in the region 103 to 106 Hz generated by

cometary pick up ions and photoelectrons are present in Comets Giacobini-

Zinner and Halley upto distances of about 2×106 km from the nucleus. The

Comet Halley emission was about an order of magnitude more intense than

as seen from Comet Giacobini-Zinner. Some of the wave modes that has

been seen in comets are shown in Fig. 10.10. They show the electron plasma

oscillations (EPO), the ion acoustic waves, electromagnetic whistlers and

lower hybrid resonance (LHR) waves.

An indication of the presence of strong turbulence even upto distances

∼ 2 × 106 km from the nucleus came from the detection of strong wave

activity. In addition to high level plasma activity, large-amplitude magnetic

field variation were observed. The turbulence measured by Vega and Giotto

Spacecrafts of Comet Halley showed that the magnetic field fluctuations

were smaller compared to that observed in Comet Giacobini-Zinner. There

is a clear correlation between the magnetic field variations and solar wind

plasma measurements which suggest the waves are fast-mode magnetosonic

waves. The observed magnetic field turbulence spectrum follows a power

law spectrum with an index of about 2, which is sufficiently different from

the fully developed Kolmogorov spectrum with a spectral index of 5/3. This

indicates that the cascade has not progressed far enough.

The detection of strong turbulence has led to several theoretical inves-

tigations. Both analytical treatments and numerical simulations have been

applied to interpret these results. They could arise due to the presence of

cyclotron harmonic emission, which would add power at the higher frequen-

cies; or if ions have a finite temperature, it will shift the wave frequencies to

higher values or due to non-linear effect of the waves and so on. It is quite

possible that several processes could lead to a smooth power law spectrum.

It is found that resonant wave-particle interactions and ion-shell formation

can lead to a wave power law spectrum with an index of around 2.0.

Though a large number of substructures have been identified in the in


Ion Tails 321

Fig. 10.10 Several wave modes seen in Comet Giacobini-Zinner in the upstream and

down stream are shown. This includes electron plasma oscillations (EPO), ion acousticwaves, electromagnetic whistler modes and lower hybrid resonance (LHR) waves (Scarf

F.L. et al. 1987. Astr. Ap. 187, 109).

situ measurements, the mechanism of formation of such structures is not

clear.

10.5. Acceleration of Cometary Ions

The possibility that cometary ions could be accelerated to high ener-

gies in the coma due to the turbulent plasma environment came from the

analogy with the diffuse shock acceleration at the Earth’s bow shock. The

existence of such accelerated ions in Comet Giacobini-Zinner came from

the observations carried out with the ICE spacecraft. The measurement

showed large flux of cometary ions of about 100 keV even upto distances

∼ 106 km from the nucleus. Large anisotropy in their distribution was

also detected. The in situ measurements of Comet Halley confirmed the

presence of appreciable fluxes of energetic cometary ions in the coma, but



it extended even beyond 100 keV and upto about 0.5 MeV.

The maximum energy that a newly formed ion can have in the rest

frame of the spacecraft is given by

Emax =1

2mi(2V⊥)2 = 2miV

2sw sin2 α (10.8)

where 2V⊥ is the total velocity comprising of the drift speed V⊥ and roughly

an equal amount of gyration speed. VSW is the solar wind speed and α is

the angle between the solar wind velocity and the interplanetary magnetic

field. For Vsw ' 400 km/sec, the maximum energy Emax that water ions

can have ∼ 60 keV corresponding to α = 90. There should be a sharp cut

off in the energy distribution beyond this value. However the observations

indicating that the energy spectra extended upto ∼ 0.5 MeV clearly showed

that significant ion acceleration must have occurred. The value of α = 90

or 0 correspond to the extreme cases of velocity vector of the ions in the

direction of the solar wind (Emax) or in the opposite direction (Emin) and

should therefore lead to anisotropies in the energy distribution.

The possible mechanisms for the acceleration of ions has led to several

studies of wave-particle interactions. The cometary ions could be acceler-

ated through diffuse shock acceleration process near the bow shock. This

process is basically related to random multiple scattering of charged parti-

cles across the shock front, separating the supersonic and subsonic sides, a

form of first order Fermi acceleration. Another mechanism is the stochas-

tic acceleration process, a form of second order Fermi acceleration process.

Here the mechanism is the scattering of the particles by the waves in the

turbulent cometary plasma. In the diffuse acceleration process, the acceler-

ated ion population builds up ahead of the bow shock with a characteristic

length scale given by K/Vsw where K is the diffusion coefficient of the

medium normal to the shock. For the case of ion acceleration at the bow

shock of Comet Halley, the estimated distance is ∼ 106 km. The diffuse

shock acceleration process is more important in this region and for ener-

gies ≤ 100 keV. For energies > 100 keV and for distance > 5 × 106 km

far from the bow shock, the acceleration through stochastic process or the

second-order Fermi acceleration process is of major importance. Support

for the stochastic acceleration process also seems to come from the obser-

vations that can be fitted well with an exponential velocity distribution of

cometary ions. Other mechanisms that have been considered for ion accel-

eration process include the lower hybrid turbulence that could be generated

by the comet ion pick up process as well as by the bow shock formation

and ion cyclotron wave absorption.


Ion Tails 323

The presence of intense electron plasma oscillations and ion-acoustic

waves in the Comets Giacobini-Zinner and Halley indicate that electron

heating and acceleration should take place. Some of the processes con-

sidered for electron heating are ion acoustic wave instability, cross-field

streaming instabilities, lower hybrid wave acceleration and so on.

10.6. Large Scale Structures

Different kinds of plasma processes must be taking place in the ion

tail of comets. This is evident from the photographs of bright comets

which show a wide variety of unusual large scale dynamical phenomena.

A wide variety of plasma tails can be seen from the Atlas of Cometary

Forms. Many of these interesting phenomena appear to be controlled to

a large extent by the microscopic plasma physical processes arising out of

the comet-solar wind interactions. There are many other features which

arise due to the dominance of the solar wind. Therefore any change in the

solar wind property in a small or on a large scale is reflected directly in

the behaviour of the cometary plasma. Some of the characteristic features

arising out of these processes are given in Table 10.2. It should be noted

that most of the observed features are transient in nature. Since the ion

tail is basically a plasma column of large extent and is also bright, it gives

a splendid opportunity for the study of spatial and temporal variations. It

can also help in distinguishing the phenomena connected with the processes

taking place inside the tail like instabilities etc. from those introduced by

the fluctuations in the solar wind.

Table 10.2 Plasma process and the possible associated

events.

Physical process Lead to observed features

Change in the solar Changes the orientation ofwind conditions the ion tail, bends the tail

Development of Enhancement of ionization,instabilities flares, condensations, filaments,

rays, helices, waves, etc.

Magnetic field lines Disconnected tailsdisconnected



10.6.1. Tail rays or streamers

The filametary structure in the ion tails of comets is quite common and

is also very conspicuous (Fig. 1.18). The shape of these streamers appears

to show the importance of magnetic fields. Since the CO+ ions present

in the ion tail follow the direction of the magnetic field, the streamers are

made visible through CO+ emission. These rays going in the anti-solar

direction grow with time and can extend from smaller lengths of about 103

km to longer lengths of about 106 km. The rays are distributed almost

symmetrically around the comet tail axis. They also seem to turn towards

the main tail axis. The time scale for coalescing process is around 10 to

15 hours. Since the time of formation of a new streamer is about one hour,

usually streamers as many as 20 to 25 can be seen at any given time. These

streamers do not interact with each other. They appear to originate from

a small volume very close to the nucleus, at a distance of the order of

103 km from the nucleus. In addition to seeing the streamers originating

from the coma, streamers starting from the tail have also been noticed.

Since these rays do not originate from the ionized region, the mechanism

of formation of these rays has to be different. Two possibilities have been

suggested either they are produced by nonlinear compressional waves or

the cometary ions trace the magnetic field lines. Observations appear to

show the first explanation to be unlikely although there are problems with

the second explanation. The ion stream could also arise due to the effect

of magnetic sector boundaries in the IMF (Sec. 10.6.6) The IMF following

the sector boundary reconnects at the front side of the ionosphere. This

can lead to ion streamers. In the coma, in addition to magnetic lines

of force, neutral sheets or surfaces also curl around and their behaviour

will be similar to the observed reversal of directions in the interplanetary

medium. It has been suggested that the streamers represent the enhanced

plasma which is confined to these neutral tubes or surfaces. These ions are

concentrated in the neutral zones as a result of the field gradients on either

side of the neutral regions. Also these neutral regions prevent the field lines

of opposite polarity merging with each other leading to annihilation. These

neutral tubes are nearly cylindrical, centred on the tail axis and when seen

edge-on, they appear as rays. They appear symmetrical about the tail

axis and this is consistent with the observations. It is quite possible that

strong shock waves in the sunward direction can generate keV electrons

and energetic protons. These high energy electrons streaming along the

magnetic field lines can result in high ionization in a short time scale which


Ion Tails 325

can give rise to streamers.

10.6.2. Knots or condensations

The plasma tails are filled with structures such as bright knots or con-

densations which are basically ion concentrations. With the use of time

sequence photographs, it is possible to follow these knots as a function of

time for a period of a few hours to a day. From such measurements, the ve-

locity and acceleration can be determined. Their velocities lie in the region

of around 20 km/sec near the head to around 250 km/sec at distances far

from the head. This corresponds to a value of (1 - µ) of around 100 with

wide variation. Several attempts have been carried out for an understand-

ing of these results. In particular whether the observed motions are the

mass motions or some form of disturbance traveling down the tail. Even

though the collision mean free path for the particles is large, it is still a

reasonable approximation to treat the interaction between the solar wind

and the cometary plasma as a fluid-like interaction. Based on such a hy-

pothesis, several other mechanisms for the observed acceleration have been

suggested. One suggested process is that the acceleration will result when

the plasma is squeezed out of the flux tubes by the magnetic pressure gra-

dients. Another point of view is that the enhancement of the momentum

transfer takes place due to some form of instability setting in the plasma. It

is not clear at the present time whether the observed motions are the mass

motions or some form of disturbance propagating down the tail. Doppler

measurements carried out in the near-nucleus region of plasma tail indicate

speeds of around 30 km/sec and extend to distances of around 3 × 105 km.

The derived acceleration is found to increase with distance. Therefore it is

likely both mass motion and waves are present in the plasma tail of comets.

10.6.3. Oscillatory structure

The photograph of Comet Kohoutek taken on 13 January 1974 when it

was at r = 0.58 AU and ∆ = 0.81 AU, clearly showed the presence of a

wavy structure in the ion tail, moving approximately at a speed of about

235 km/sec. (Fig. 1.16). The feature appeared to be a helical structure.

The radius and the wavelength of the structures are about 2.3×105 km and

1.4 × 106 km respectively. The wavy structure had an extension of about

3.6 × 106 km and was about 1.6 × 107 km away from the nucleus. These

oscillations have been interpreted in terms of “kink instability” based on



the studies of the Earth’s geomagnetic tail. The theoretical work developed

for the Earth’s geomagnetic tail has been successfully applied to the comet

tails. In brief, an azimuthal component of the magnetic field is produced, in

a cylinder of plasma when a current flows through the longitudinal magnetic

field. This finally gives rise to a helical field. The phase speed of the helical

kink turns out to be simply the Alfven speed of the tail plasma, which is

given by

C ≈ B

(4πρ)1/2(10.9)

where B is the magnetic field and ρ is the mass density. For a number

density of CO+ ions of 10 to 103/cm3, the magnetic field lies in the range

of

100γ ≤ B ≤ 1000γ.

10.6.4. “Swan-like” feature

Another large scale dynamical structure in the tail of Comet Kohoutek

can be seen in the photograph of 11 January, 1974 (Fig. 1.17). This is

generally termed as “Swan-like” feature because of its appearance. The

size of the feature is estimated to be about 5 × 106 km and is situated

at a distance of about 15 × 106 km from the nucleus. The feature was

found to move away from the nucleus with an apparent speed of about 250

km/sec. The Swan feature is not a mass flow but rather a propagating

hydromagnetic wave phenomenon. In fact it could be the kink instability

in an advanced stage of growth.

10.6.5. Bend in the tail

The changes in the solar wind conditions can reflect directly in the

observations of ion tails. This was established for the first time, based on

the photographic observations of Comet Bennett. It was shown that the

kinks observed in the tail of this comet were correlated with the solar wind

events as observed from the satellite measurements.

A striking example of this type can be seen from the photograph of

Comet Kohoutek taken on 20 January 1974, where a large bend can be

seen. This is shown in Fig. 10.11. It is interesting to investigate whether

this bend is correlated with the solar wind properties for that date. The

properties of the solar wind are obtained by the observations made by


Ion Tails 327

Fig. 10.11 Photograph of Comet Kohoutex taken on January 20, 1974 which shows the

large scale bending of the tail. (Brandt, J.C. and Chapman, R.C. 1981. Introduction toComets. Cambridge: Cambridge Univ. Press). Illustration credited to Joint Observatory

of Cometary Research, NSA.

the Earth orbiting Satellite IMP-8. These observations have been used in

conjunction with the theory to predict the nature and structure of the ion

tail projected on to the sky for the day of the observation. There was a good

agreement between the expected and the observed comet tail configurations.

These results clearly showed that the observed large-scale tail curvature

coincided with the compression region of a high speed solar wind stream.

Generally a high speed solar wind stream causes enhanced geomagnetic

activity which usually shows up in the geomagnetic records. A geomagnetic

storm was actually detected on January 24 to 27, corresponding to the

Comet Kohoutek observations. Figure 10.12 shows schematically the tail

configuration and the bend. The direction of the solar wind is shown by

the arrows. It can be seen that in region A, the tail has the same direction

as the direction of the solar wind. This means that the tail has taken the

equilibrium position. In region B, the flow of the solar wind is across the

tail and has not yet adjusted to the direction of the solar wind. On the

other hand the segment C is well aligned with the head of the comet, as the

tail has not yet been affected by the solar wind. This is a striking example



Fig. 10.12 Schematic representation of Fig. 10.11. The left portion shows overall kinkand the right portion shows some details. The inferred solar wind direction is shown by

arrows. (Adapted from Jockers, K. 1981. Icarus, 47, 397).

which clearly shows that the comet tails are very effective for monitoring

the changing conditions of the interplanetary plasma.

10.6.6. Disconnection events

Another interesting and characteristic feature quite often seen in the

plasma tail of comets is the disconnection events which have been noted

since early times. The photograph of Comet Morehouse shows the abrupt

disconnection of the tail as shown in Fig. 10.13. This is a common prop-

erty of all comets. The disconnection events appear to be cyclic in the

sense that after a tail gets disconnected a new tail appears to form and this

process seems to repeat itself. A possible mechanism for such phenomena


Ion Tails 329

Fig. 10.13 Photograph of Comet Morehouse showing a tail disconnection event. Left

and right photographs refer to September 30, 1908 and October 1, 1908 respectively.

(Niedner, M.B. and Brandt, J.C. 1979. Ap. J. 223, 671).

can be understood from Fig. 10.14. The explanation is essentially related

to the presence of magnetic sector boundaries in the interplanetary mag-

netic field (IMF). A sector boundary is defined as the boundary between a

region in which the field lines are directed primarily towards the Sun (in the

ecliptic plane) and a region in which the field lines are directed primarily

away from the Sun (in the ecliptic plane). The two regions with opposite

magnetic field polarity are separated by a thin layer of the heliospheric cur-

rent sheet (HCS). When the comet ionosphere crosses the magnetic sector

boundary of the IMF at which the magnetic field direction reverse by 180o

i.e. HCS, it disconnects the existing tail and it will move in the anti-solar

direction. A new tail will then form with the reconnection of the field lines

with the opposite field orientation until a new magnetic sector comes along.

The process is repeated. Therefore if a bright comet is observed for a long

time, it is possible to observe many events as described above. In the re-

connection phase, the magnetic energy may be dissipated in the form of

an increase in brightness or ionization or flares etc., and it may show up

in the observations. There is some observational evidence to suggest that

these things do happen. Comet Halley showed a disconnection event on

March 8.4, 1986. The presence of magnetic sector boundary at this time

was verified from observations of Vega 1 and Vega 2 spacecrafts. Comet

Hyakutake also showed a spectacular disconnection event on March 24.6,



Fig. 10.14 Process of formation of disconnected plasma tail of a comet due to the pas-

sage of an interplanetary sector boundary. Tail ways are shown by the shaded portions.

(Niedner, M.B. and Brandt, J.C. 1979. op. cit.).

1996. This was also in agreement with the calculated position of the mag-

netic sector boundary. Therefore all the observations are in agreement with

the magnetic sector boundary reconnection model.

10.7. X-rays

X-ray emission is generally associated with a high temperature plasma of

about 106 K or so. Therefore the detection of X-rays from Comet Hyakutake

with ROSAT satellite in 1996 was a great surprise. The motivation to

look for X-rays in Comet Hyakutake appear to have come from a study

of the possible production of X-rays as a result of high speed collision

between interplanetary dust particles and the dust particles in the comet.


Ion Tails 331

Comet Hyakutake was considered to be a favourable comet for the possible

detection of X-rays as the comet was very bright and came very close to the

earth. Strong X-ray emission was indeed detected for the first time from this

comet. Following the detection of X-ray from Comet Hyakutake, ROSAT

archival data base was searched for the possible presence of other comets

with X-ray emission and several comets were found. In recent times, many

more comets have been detected in the X-ray region from X-ray satellites.

Therefore X-ray emission can be considered to be a common feature of all

comets.

Several characteristic features associated with X-ray emission from

comets have been seen. X-ray images appears to show similar spatial mor-

phologies, which is cresent shaped extending upto about 105 km with the

brighter peak displaced toward the Sun (Fig. 10.15). The X-ray emission

Fig. 10.15 Shows the images of Comet Hyakutake taken on 26–28 March 1996. The

first two are the X-ray images taken with ROSAT. The third image is in the visiblelight with X-ray emission contours superimposed. The mark “+” denotes the location

of the nucleus and the direction of the Sun is towards the right (Lisse, C.M. et al. 1996.

Science, 274, 205).

is confined to the cometary coma. In the case of Comet Hyakutake the

location of the emission peak from the nucleus was about 2 × 104 km.

These give an indication of the possible encounter of solar wind with the

extended cometary atmosphere. The X-ray emission was not associated

with extended dust or plasma tails. The X-ray emission was also found to

be variable. The observed total luminosity of Comet Hyakutake was about

4 × 1015 ergs/sec for an aperture radius at the comet of 1.2 × 105 km. In

the case of Comet Levy the observed soft X-ray luminosity in the energy

range 0.2 to 0.5 keV was about 2 × 1016 ergs/sec. Observations of sev-

eral comets have shown that total X-ray luminosity correlates with the gas

production rate. The time variation of X-ray emission show a correlation



with solar wind proton flux and oxygen ion flux, but no correlation with

solar X-ray flux. Longtime monitoring of rapid changes in gas production

and the observed X-ray flux in comets (such as an outburst) has shown a

direct link betwen the two. Earlier X-ray observations carried out at low

energy resolution could not resolve the possible presence of emission lines.

However with the availability of higher resolution instruments, the spec-

trum was found to be dominated by emission lines from highly charged C,

N and O and not by continuum (Fig. 10.16).

Fig. 10.16 X-ray spectrum of Comet C/1999S4 (LINEAR) observed with Chandra X-ray Observatory (crosses). The six-line best-fit model spectrum is shown as solid line

(Lisse, C.M. et al. 2001. Science 292, 1343).

Any physical mechanism proposed to explain the observed X-ray emis-

sion in comets has to be consistent with all the observed characteristics of

this emission. These include thermal bremsstrahlang associated with col-

lision of solar wind electrons with neutral gas or dust, scattering of solar

X-rays by cometary small dust particles, charge exchange of solar wind

heavy ions with cometary neutrals, K-shell ionization of cometary neutrals

by electron collision etc.

In the Thermal bremsstrahlang mechanism, the electrons of solar wind


Ion Tails 333

are deflected from the nuclei of atoms on collison and give rise to continuum

radiation. For such a process the electron energies have to be larger than

100 eV (i.e. T>106 K). In the K-shell mechanism the collision between

a fast solar wind electron and an atom can remove the orbital electron

from the inner shell of the atom. However in these mechanisms the ex-

pected luminosity are smaller by a factor of around 100 to 1000 compared

to observations. This is due to the fact that flux of high-energy solar wind

electrons near comets are not enough. Also X-ray emission has been ob-

served upto larger distance from the nucleus (∼ 105 to 106 km) beyond the

bow shock and the energies of solar wind electrons at these distances is only

around 10 eV. In addition the presence of multiple emission lines indicate

non-viability of continuum or K-shell line models.

There are also difficulties associated with mechanisms involving dust

models. The scattering of solar X-ray radiation by cometary dust particles

of sizes ∼ 1 µm in size cannot produce the observed X-ray luminosities.

This problem can be overcome by involving the presence of small dust par-

ticles of size of the order of the wavelength of observed X-ray emission,

∼ 10–100 A, which can scatter the solar X-ray radiation efficiently. The

abundance of such dust particles in comets is not known. However there

was some indication of the presence of such dust particles in Comet Halley

from Vega flyby spacecraft. Other problems associated with dust particles

are X-ray emission correlates with gas production rate, time variation of

X-ray emission correlate with solar wind ion flux etc. In view of these con-

siderations the charge exchange collisions of highly charged solar wind ions

with cometary neutral species and the collison of high energy solar wind

electrons with cometary neutrals are plausible processes. Charge exchange

mechanism produce lines while electron neutral collision produce a contin-

uum. In both of these mechanisms, neutral species in the cometary coma

is the target.

The heavier minor species in the solar wind exist in highly charged

states such as O7+, O6+, N6+, C5+, Ne8+, Si9+ etc. In the charge exchange

mechanism, these highly charged ions charge exchange with cometary neu-

tral species and produce ions in an highly excited state. A typical example

of such a process is,

O6+ + H2O→ O5+ ∗ + H2O+

Here O6+ ion in collision with an abundant neutral molecule such as H2O

produce O5+∗ in an excited state denoted as ? and H2O+ ion. Other

abundant neutral molecules can be OH, CO etc. The excited state of the



ion O5+ ∗ decay spontaneously giving rise to photons in the X-ray region.

It is possible to make an estimate of the X-ray power density for the case

of a single collision with the solar wind ion in the coma from a simple

expression,

Px = αnswuswnn. (10.10)

where nsw, usw and nn represent solar wind proton density, solar wind speed

and density of the neutral species respectively. The parameter α includes all

other quantities partaining to solar wind, atomic and molecular details. The

variation of neutral density of the species with r in the coma for a spherical

symmetric case is given by nn = Q/(4πr2un) where Q is the production

rate of the specie. Integration of equation (10.10) over the observed volume

of the neutral species result in the X-ray luminosity which is roughly within

a factor of 2 to 3 of the observed X-ray luminosity. The X-ray luminosity

is also proportional to the gas production rate which is in agreement with

the observations.

The cometary gas density is high close to the nucleus and hence col-

lisions are frequent. This results in solar wind ions being converted to a

low charged state ion and therefore X-rays cannot be excited. This could

explain the observed crescent - shaped X-ray emission. The model spectra

based on charge exchange mechanism were in good agreement with the low

resolution cometary X-ray emission.

For a better understanding of the X-ray emission including the observed

X-ray line emissions, it is necessary to invoke detailed Magneto Hydro-

dynamic models (MHD) for the interaction of the solar wind with the

cometary gas which allows variation in plasma density, temperature, ve-

locity etc. The variation in the neutral density of species such as H2O,

CO2, CO, OH etc. can be included through Haser’s model. The details of

the charge-exchange mechanism such as, the abundance of various ions and

its variation and the cascading process leading finally to X-ray emission

should be an integral part of this study. Such sophisticated models have

been carried out to calcuate the expected X-ray emision around a comet.

The simulated X-ray emission are found to be in good agreement with the

observed images (i.e. isophotes). The X-ray line emission spectra can also

be calculated. These are in good agreement with the observed spectra

(Fig. 10.16). Therefore the observed characteristics of X-ray emission from

various comets can be explained based on charge-exchange model.


Ion Tails 335

10.8. Summary

The existence of solar wind came from the studies of the orientation

of ion tail of comets, long before its detection by spacecrafts. The whole

problem of comet-solar wind interaction is a highly complex one. The ex-

act nature and the location of the different discontinuities depend upon the

various physical processes that can take place during the interaction. What

happens in a particular situation depends upon the complex interaction of

the plasma, turbulence, solar activity and so on. However the large scale

structure as predicted by the MHD calculations like bow shock, ionopause,

magnetic cavity and so on have been confirmed from the in situ measure-

ments of Comets Giacobini-Zinner, Halley and Borrelly. This shows that

the mechanism of interaction between the solar wind and cometary plasma

giving rise to various structures is fairly understood at the present time.

The measurements also showed the plasma to be highly complex in showing

the presence of strong turbulence, various instabilities, waves, high energy

particles and so on. Various mechanisms have been proposed to explain

the origin of these observed features. Strong X-ray emission a is common

feature of all comets.

All the properties of the ion tail that have been discussed so far can be

classified as belonging to the “typical” comets. In these comets the ion-

tails are more or less similar and behave in a certain manner except for the

presence of certain characteristic features in the tail in some cases. The

major aim has been to understand the gross behaviour of such “typical”

comets.

Problems

1. Take for the velocity of the solar wind v = 450 km/sec and the proton

density n = 5/cm3. Calculate the magnetic field required to counteract

the solar wind pressure.

2. Does the magnetic field exist in comets? Can you suggest some method

to probe the magnetic field associated with a comet?

3. Calculate the Alfven speed in the tail of comets.

4. Discuss the relative importance between cometary plasma and labora-

tory plasma



References

The evidence for the existence of solar wind is discussed in this paper

1. Biermann, L. 1951. Zs. f. Astrophysik 29, 274.

For a complete theoretical discussion of the solar wind, refer to

2. Parker, E.N. 1963. Interplanetary Dynamical Processes. New York:

Interscience Publishers.

The phenomena of dynamical aberration is discussed in the following pa-

pers.

3. Belton, M.J.S. and Brandt, J.C. 1966. Ap. J. Suppl. 13, 125.

4. Brandt, J.C. and Mendis, D.A. 1979. In Solar System Plasma Physics,

Vol. II. eds. C. F. Kennel, L.J. Lanzerotti and E.N. Parker. Amster-

dam: North-Holland Publishing Company p. 253.

5. Hoffmeister, C. 1943. Zs. f. Astrophysik 23, 265.

The Wind-Sock theory for the ion tail is discussed here.

6. Brandt, J.C. and Rothe, E.D. 1976. In The Study of Comets, eds. B.

Donn. M. Mumma, W. Jackson, M.F. A’Hearn and R. Harrington,

NASA SP 393, Washington D.C. p. 878.

The basic idea of the solar wind interaction is given in this paper.

7. Alfven, H. 1957. Tellus 9, 92.

The theoretical model of the Comet-Solar Wind interaction is from

8. Wegmann, R. et al. 1987. Astr. Ap., 187, 339.

Later work

9. Gombosi, T.I., De Zeeuw, D.L. and Haberli, A.M. 1996. J. Geophys.

Res., 101, 15, 233.

10. Ip, W. -H. 2005. In Comets II, eds. M.C. Festou, H.U. Keller and H.A.

Weaver, Univ. Arizona Press, Tucson, p. 605.

The confinement of ions very close to the nucleus was discussed by

11. Wurm, K. 1963. In The Moon, Meteorites and Comets, eds. B.M. Mid-

dlehurst and G.P. Kuiper, Chicago: The University of Chicago Press

p. 575.

The following atlas gives cometary plasma tails of various forms.

12. Rahe, J. Donn, B and Wurm, K. 1969. Atlas of Cometary Forms,

NASA SP-198, Washington D.C. GPO.

A good discussion of the in situ plasma measurements can be found in the

following papers:

13. Flammer, K.R. 1991. In Comets in the Post-Halley Era. eds. R.L.

Newburn, Jr. et al. Kluwer Academic Publishers. p. 1125.


Ion Tails 337

14. Tsurutani, B.T. 1991. In Comets in the Post-Halley Era. eds. R.L.

Newburn, Jr. et al. Kluwer Academic Publishers. p. 1171.

15. Ip,-W.H. and Axford, W.I. 1990. In Physics and Chemistry of Comets,

Springer Verlag, p. 172.

The electron temperature is discussed in the paper

16. Eberhardt, P. and Krankowsky, D. 1995. Astron. Astrophys., 295,

795.

Model for Disconnection event was proposed in the following paper

17. Niedner, M.A. and Brandt, J.C. 1978. Ap. J., 123, 655.

18. Brandt, J.C., Caputo, F.M., Hoeksema, J.J. et al. 1999. Icarus, 137,

69.

Possibility of detection of X-ray from comets is discussed in the paper

19. Ibadov, S. 1996. Adv. Space Res., 17:(12), 93.

Discovery of X-rays from Comet Hyakutake is in the following paper

20. Lisse, C.M. et al. 1996. Science, 274, 205.

For later work, refer to the following papers

21. Cravens, J.E. 2002. Science, 296, 1042.

22. Lisse, C.M., Cravens, T.E. and Dennerl, K. 2005. In Comets, eds. M.C.

Festou, H.U. Keller and H.A. Weaver, Univ. Arizona Press, Tucson,

p. 631.

23. Wickramsinghe, N.C. and Hoyle, F. 1996. Astrophys. Space Sci., 239,

121.


CHAPTER 11

Nucleus

Though the nucleus is responsible for most of the observed phenomena,

its nature is the least understood. As the nucleus cannot be observed

directly from the Earth because of its small size, the information with

regard to their possible nature, structure and composition has to come

from indirect means. The flybys to Comets Halley, Tempel 1, Wild 2 and

Borrelly have given new insights into this area.

11.1. Morphology

Several surprising results came out on the surface characteristics of the

nuclei of comets from spacecraft flybys. The spacecrafts Giotto, Vega 1

and Vega 2 were the first to take the images of the nucleus of the Comet

Halley. Based on different aspects seen by the three spacecrafts, the actual

three dimensional shape of the nucleus was constructed. It clearly showed

the nucleus is not spherical in shape, but highly irregular. It resembles

more like a ‘peanut’ or a ‘potato’. The surface has many features such as

valleys, hills and craters. The nucleus was also found to be large as well

as quite dark in nature. The salient feature was that the gas emission was

not uniform over the entire surface of the nucleus but comes out as discrete

jets. Due to rotation of the nucleus the jets will become active only when

they are exposed to the solar radiation. The jets cover around 10% of the

surface of the nucleus.

The close-by images of Comet Borrelly by Deep Space 1 mission showed

the nucleus to be shaped like a ‘foot print’ (Fig. 11.1). The images also

showed jets as was seen in Comet Halley. The terrain appeared to be

highly complex with some craters. They also show bumps, troughs and

other features. The nucleus of Comet Wild 2 observed in stardust mission

339



was found to be oblate. It contains depressions ranging in size up to about

2 km across. Some of the depressions are not circular and have complex

structures. They could have been formed by impact, sublimation, ablation

or a combination of all of these processes.

The images of Comet Tempel 1 taken by Deep Impact showed the pres-

ence of circular depressions and terrains. One dramatic feature seen was

the presence of layers over the entire surface of the nucleus of the comet

(see Fig. 11.8). Such layers are also found in Comets Borrelly and Wild 2

based on the images taken by the earlier spacecrafts.

The Comets Borrelly, Wild 2 and Tempel 1 belong to the same group of

Comets, namely Jupiter family, but their appearances are vastly different,

as can be seen from Fig. 11.1. This could either be connected with the

processes that could have taken place on the surface itself or could possibly

be due to different internal histories.

11.2. Theory of Vapourization

When the solar radiation impinges on the surface of the nucleus, part

of the energy is absorbed and part of it is reflected. The reflected part

depends upon the albedo of the nucleus. The absorbed energy gives rise to

the surface temperature which sublimates the material and also gives rise

to the thermal emission. The absorbed energy also depends upon the angle

between the incident radiation and the normal to the surface.

The amount of energy radiated from the surface is dependent upon the

temperature, which in turn is related to the chemical composition and the

structural properties of the nucleus. The sublimation of the gases depends

upon the latent heat and its variation with temperature. The penetration of

the heat flow inside the nucleus also depends upon the conducting properties

of the material. The presence of dust in the coma can also have an effect

on the vapourization of the volatile components. Therefore the amount of

volatile constituents that come out of the surface of the nucleus is a very

complicated function involving many unknown factors. Hence, until we

have better ideas with regard to the above processes, one has to resort to

simplifying assumptions in writing down the equation for the energy flow.

The temperature of the nucleus is determined by the balance between

the amount of absorbed and the emitted energy. The energy balance equa-

tion for the simple model of a spherical nucleus which takes into account

the thermal re-radiation and the heat used to transform the vapourizing


Nucleus 341

Fig. 11.1 Comparison of the images of nucleus of Comets Tempel 1 (upper right), Wild 2(upper left) and Borrelly (bottom) taken from spacecrafts. The morphological structure

of each of these comets is entirely different (Thomas, P.C. et al. 2007. Icarus, 187, 4).

ice into gas for the steady state condition can therefore be written as

F(1−Av)r2

cos z = (1−AIR)σT 4 + Z(T )L(T ). (11.1)

Here F is the solar radiation at 1AU. Av and AIR are the bond albedo

of the nucleus in the visible and infrared region and r is the heliocentric

distance. Z(T ) and L(T ) represent the vapourization rate of the gas in

molecule/cm2/sec and the latent heat for sublimation in ergs/molecule. T



is the surface temperature and z is the solar zenith distance. The vapour-

ization rate Z(T ) can be deduced by relating it to the equilibrium vapour

pressure of ice. Under the equilibrium conditions the number of molecules

coming out of the nuclear surface will be balanced by the number hitting the

surface. The number of molecules hitting the surface for such a situation

is equal to 14Nv. Therefore

Z(T ) =1

4Nv (11.2)

where N is the gas density and v is the mean speed for the Maxwellian

distribution of velocities which is given by

v =

(8kT )

πm

)1/2

(11.3)

Here m is the average molecular weight of the gas. Using the above relation

for v and with the use of the relation P = NkT,Z(T ) can be written as

Z(T or r) =P

(2πmkT )1/2. (11.4)

The total production rate of the molecule is given by

Q(r) = S < Z(r) >

where < Z(r) > represents the production rate averaged over the solar

zenith distance z and S is the surface area of emission. The integration over

z can be avoided under some simplifying assumption. For a non-rotating

nucleus of radius rn, the cross section for the absorption of solar energy

is πr2n, while for a fast rotating nucleus the incident energy is uniformly

distributed over the entire surface 4πr2n. The corresponding value of <

cos z > is 1 and 1/4 for the slow rotator and fast rotator models respectively.

The equations show that even the very simple energy balance equation

requires the specification of the six parameters, namely Av, AIR, L, P , η

and S. Here η represents the geometrical factor related to cos z. Since

it is a multivariable problem, it is necessary to make some assumptions

to limit the number of free parameters. This depends upon the problem

under investigation. It then fixes certain parameters and others are derived

from comparing with observations. This procedure is just a method of

simplifying the problem and therefore the results have to be weighted in

terms of the assumptions made in the problem.

Since H2O is the dominant component of the nucleus of a comet, the

values for L and P corresponding to H2O are generally used. The vapour


Nucleus 343

pressure can also be calculated from Clasius-Clapeyron equation

P = Pr exp

[L(T )

RT

(1

Tr− 1

T

)]where Tr and Pr are the reference point. The vapour pressure in general

is very sensitive to temperature. Therefore, the general variation of the

production rate of H2O and OH as a function of the heliocentric distance

should reveal something about the vapourization of ice from the nucleus

of a comet and should in turn give information about the overall nature

of the nucleus of a comet. A simultaneous solution of Eqs. (11.1) and

(11.4) iteratively will give the unknown Z(T ) and T , for assumed values of

the parameters that enter in the above two equations. The results for the

variation of Z as a function of the heliocentric distance for different cases are

shown in Fig. 11.2. The curves depend upon the values of Av and AIR. The

Fig. 11.2 The vapourization rate Z, for various snows as a function of the heliocentricdistance. The distance after which the vapourization becomes negligible is denoted asr0. (Delsemme, A.H. 1982. In Comets, ed. Wilkening, L.L. Tucson: Univ. ArizonaPress, p. 85).

curves are almost the same if Av = AIR. For unequal values of Av and AIRthe maximum difference in the results arises in the shift of the curves in log

r by about ±0.2. Figure 11.2 shows r−2 dependence in the beginning and

then shows a rapid fall in the sublimation rate of water for distances between



2 and 3 AU. The calculated behaviour of the curve is basically related to

the steep dependence of vapour pressure on temperature. The sublimation

rate is nearly proportional to the impinging energy if the temperature is

greater than a certain critical value. This is the case for small values of r

and hence Z varies as r−2. For distances greater than a certain value, the

sublimation rate falls down steeply with temperature and this gives rise to

the observed effects. The cometary observations also show a similar cut-off

around these distances, after which they are not generally seen. This effect

is particularly so in the case of short period comets where the cut-off is

around r ∼ 3 AU. These observations are in striking agreement with the

expected curve for the sublimation of H2O. Therefore the observations by

itself seem to indicate that H2O may be the dominant constituent of the

nucleus. In addition, the vapourization theory gives the total production

rate of rough order with the observed rates of several comets where the

water controls the sublimation process.

Based on the above model, it is rather difficult to explain high volatiles

that has been seen in some comets at large heliocentric distances of around

3 AU, where it is unlikely that water will evaporate. This led to the sugges-

tion of Clathrate Hydrate model. The clathrates are basically loose crystal

lattices of H2O in which molecules like CH4, CO2, etc are loosely packed.

These molecules are not chemically bound but they are bound by weak Van

der Waals forces. The cut-off distances for such types of clathrates lie in the

range of about 5 AU or so. Therefore the comets which have been observed

beyond r>∼ 3 AU are likely to arise out of this structured ice material. This

was a significant step in the understanding of the cometary nuclei. In this

connection it is interesting to note that OH has been detected in Comet

Halley at r = 4.9 AU with logQ(OH) ∼ 29 and in Comet Hale-Bopp at

r = 5.13 AU with log Q(OH) ∼ 27.19.

In the derived production rate of various volatiles shown in Fig. 11.2

the value for the albedo for the nucleus of pv ∼ 0.7 was used. However ex-

tensive observations carried out on Comet Halley followed by other comets

indicated that the albedo of the nucleus is quite low and is around 0.04.

The effect of this low albedo for the nucleus on the production rate of H2O

show a change in slope in the region of around r = 4 to 6 AU for H2O

production. Beyond this distance, production rate of H2O decreases but is

still not negligible. Therefore there is still some water coming out of the

nucleus at large distances.

Although clathrate hydrate model was in conformity with earlier

cometary observations, later laboratory studies indicated a major difficulty


Nucleus 345

in their formation. These experimental studies showed that clathrate hy-

drates is produced under high pressure conditions and hence not applicable

to comets, which are produced at low pressure and temperature. Besides,

the abundances of the observed species are quite high and is not possible

to trap so much in the clathrate hydrate icy matrix. This has led to a

new model where water-ice is most likely to be present in amorphous form.

Amorphous ice exists at lower temperatures of around 77 K or less. The

study of solar nebular models have shown that these conditions exist at

distances > 7 AU. Hence interior of comets is most likely to have amor-

phous ice, as comets were formed under low temperature and low pressure

conditions.

Amorphous H2O - ice can trap large amount of volatiles and when

heated, their trapped volatiles are released. For temperature starting

around 90 K, amorphous form of ice changes over to crystalline form with

the release of energy. This corresponds to distances of around 8 to 20 AU

where large amount of traped gases may be released. In this connection

it is interersting to note that volatile species of various kinds were seen in

abundance from Comet Hale-Bopp beginning at distances of around 7 AU.

These are most likely to be produced by amorphous to crystalline H2O - ice

phase transitions rather than produced by sublimation due to solar heating.

The early results of the vapourization of H2O from the nucleus had

met with reasonable success in explaining the gross observed nature of

the light curves of various comets. However observations have shown that

the light curve for pre-and post-perihelion passages are not symmetrical

in shape and gas comes out in jet form at some discrete active regions

on the nucleus. The active areas are hardly a few percent of the surface

area of the comets. The use of constant values of Av, AIR, L and P in

modelling means that the material property of the nucleus is the same for

all comets. Several attempts have been made to refine the models that take

into account factors like, thermal properties of the surface layers, rotation

period of nucleus, the effect of diurnal heating and cooling individually or

in combination. Though these studies have met with partial success, these

simple models have been quite useful for an understanding of the general

trend as well as qualitative study of Comets.

Refined Models

The striking feature of a comet is their varied features along the orbit

of a comet. Hence complicated models have to be considered which take

into account several processes simultaneously. Some of the processes con-



sidered are, diurnal temperature variation over the surface of the nucleus,

heat conduction to the interior through the entire nucleus, possibility of

the outer dust mantle enveloping the icy nucleus, cometary ice being in

amorphous state which can lead to crystallization, poracity, radioactive

heating from the short lived isotope 26Al. Since half life of 26Al is less

than about 0.7 million years, it would have decayed during the early stages

of the solar system, within few million years and thus providing a heat

source. The initial mass fraction of 26Al in the solar nebula ≈ 7 × 10−7.

In comets it could be smaller on the average by an order of magnitude, as

the time of cometary formation did not exceed a few million years. The

change over from amorphous to crystalline state could provide additional

source of energy. Therefore there are three sources of energy available to

comets; namely solar radiation, radioactivity and process of crystallization.

When the comets are near the sun, solar radiation dominates the other

two sources. On the other hand, the energy source through radiactive de-

cay is important that are far from the sun. However crystallization from

amorphous ice can occur at any stage in the cometary evolution once the

threshold temperature is reached.

Heat could flow inwards or outwards from any region of the nucleus.

The effect of heat and gas diffusion in the nucleus results in chemically

differentiated layers. In particular the top layers of the nucleus should

contain mostly least volatile material followed by layers of water ice and

dust and the deeper layers should contain dust, ices and most volatiles such

as CO and CH4.

The built up pressure from the gases released from the interior of the icy

nucleus, forces the gas to flow towards the surface. The flow of the gas drags

the dust particles from the icy solid matrix. It is possible that the built in

pressure could be high enough to make cracks in the porous matrix leading

to out-gassing. Large size particles reaching near the surface may lead

to formation of a dust mantle resulting in diminishing cometary activity.

Therefore the observed activity of a comet is a resultant effect of several

physical processes involving the structure of the nucleus, heat conduction

and energy sources. Several attempts have been made to study the nuclei

of comets taking into account all these details. The model considered is

a porous agglomerates made up of amorphous ice, dust and volatiles such

as CO, CO2, HCN, N2 etc. Such a model results in explaining the overall

observed activities of comets. The thermal evolution and the resulting

pattern in the activities from the nucleus of a fragile and porous comet

leads to the structured nature of the nucleus of comet as shown in Fig. 11.3.


Nucleus 347

This is quite different from that of a solid body of a nucleus made up of ice

sublimating from the surface due to solar heating as in the earlier models.

Fig. 11.3 Schematic representation of the layered structure of a cometary nucleus with

pristine composition deep inside and the modified layers arising as a result of variousphysical processes taking place (Prialnik, D. 1999. Earth, Moon and Planets, 77, 223).

The analytical expression for the vapourization of water from an icy

nucleus can be written as

Z = Zoα

(r

ro

)−m [1 +

(r

ro

)n]−k= 7Zog(r) g(r) = α

(r

ro

)−m [1 +

(r

ro

)n]−k(11.5)

where ro = 2.808 au, m = 2.15, n = 5.093 and k = 4.6142. The value of α is

chosen such that z = zo for r = 1 AU, which gives α = 0.111262. As will be

discussed later, the vapourization from an icy type of nucleus can explain

in principle the observed nongravitational forces in comets (Sec. 11.10).



11.3. Outbursts

Outbursts have been seen quite frequently in comets. It is basically re-

lated to a sudden increase in the production of volatiles from the nucleus.

Comet Schwassmann-Wachmann is well known for its large outbursts and

flares up frequently. These outbursts can brighten the comet by a factor

of 100 or more compared to the quiescent brightness. Initially, the comet

looks star-like in appearance. After the outbursts, a halo forms and even-

tually it fades away leaving behind the original nucleus. The whole process

might take around 3 to 4 weeks. The outbursts from Comet Schwassmann-

Wachmann have been seen regularly from heliocentric distances of around

5 to 7 AU. Comet Halley showed an outburst when it was at r = 14 AU.

It is rather difficult to understand such a phenomena based on the subli-

mation of H2O from the nucleus, as the production rate of H2O falls down

steeply for r >∼ 3 AU. A number of suggestions and ideas have been pro-

posed to explain the origin of these outbursts. It has been suggested that

exothermic chemical reactions involving free radicals or pockets of volatile

gas stored beneath the surface takes place until the pressure would build

up which leads finally to explosions. The phase change from amorphous

to crystalline ice at temperatures above 140 K is an exothermic process

giving about 24 cal/gm, which is a substantial amount of energy and could

trigger the outbursts. Since the phase change is critically dependent upon

the temperature, the outburst should depend on the heliocentric distance.

Therefore crystallization of amorphous ice appears to be the best mech-

anism for explaining the appearance of outbursts in comets. The three

physical processes of interest are crystallization, conduction and sublima-

tion and their relative time scales. For understanding the resultant effect

of these processes inside a porous nucleus, which are complicated in nature,

it becomes necessary to resort to numerical methods.

When the surface of the nucleus is heated due to solar heating, the heat

wave moves inwards. This results in amorphous ice to crystallize liberating

trapped gases. Part of the released gas move inwards into the colder regions

and eventually condense at different location inside the nucleus depending

upon the temperature in that region. The recondensation process results

in the release of heat which in turn affects the chemical composition of its

surroundings. During the next perihelion passage the heat wave generated

reaches these regions and is used to sublimate condensed gases rather than

the crystallization of amorphous ice. This is the process by which alter-

native layers of crystallization and amorphous ice are created. Such layers


Nucleus 349

extend from about 10 metres to about a few hundred metres below the sur-

face of the nucleus. When the heat wave from the surface reaches to some

of these layers the trapped volatile species are suddenly released leading to

pressure build-up. These gases move to the surface through pores, cracks

etc. This gives rise to an outburst. The resulting production rate of dif-

ferent volatiles could be vastly different. This could explain the observed

activities in comets (Fig. 11.4).

Fig. 11.4 The production rate of volatiles based on model calculations for Comet9P/Tempel 1 is plotted as a function of time measured from the perihelion point (Prial-

nik, D. 2005. In Asteroids, Comets and Meteors, IAU Symposium No. 229, eds, Lazzaro,

D.Ferraz-Mello, S. and Fernandez, J.A., Univ. of Cambridge Press, Cambridge, p. 153).

Model calculations have also been carried out to long time scales to un-

derstand the outburst seen in comets at large heliocentric distances. These

results indicate that such bursts are related to fine-tuning of time scales

of re-condensation of volatiles and crystallization of amorphous ice. It is

possible to attain a temperature ∼200 K in a narrow region and for a brief

period of time which can lead to sudden release of trapped gases which is

seen as an outburst. It is also possible to get repetitive outbursts if phase

transition time scale is comparable to diffusion time scale of moderately

volatile gases.



11.4. Albedo and Radius

The dimension of the nucleus of a comet cannot be obtained directly

as it cannot be resolved. The general method which is commonly used is

from the photometry of the comet at far off distances from the Sun. At

these distances the comet is more or less stellar in appearance. This comes

about due to the fact that at these distances, the temperature of the nucleus

will be so low that the sublimation of the gases from the nucleus cannot

take place. Therefore one is essentially seeing the nucleus of the comet.

Hence the observed radiation at these distances essentially comes from the

reflection of the incident solar radiation by the nucleus of comet. From

such photometric observations, the radius of the nucleus R can be obtained

from the relation

R2 = r2A−1φ(α)−1100.4[V−(Vc−5 log ∆)] (11.6)

where φ(α) is the phase function according to Lambert’s Law, A is the

albedo for the nucleus, V and Vc are the absolute magnitudes of the Sun

and the comet respectively. Equation (11.6) has been applied to several

comets using photometric magnitudes and a reasonable range of values for

the albedo of the nucleus. However, it is possible to get simultaneously the

values of radius and the albedo of the nucleus from the following simple

physical considerations.

The photometric observations of comets made at great distances from

the Sun give only the total brightness which is AS where S is the effective

cross section of the nucleus for the reflection of the solar radiation. It can be

estimated from the observed magnitudes. It is also possible to get the value

of the quantity (1−A)S from the theory of vapourization of ice-water from

the nucleus when the comet is near the Sun wherein the vapourization is

maximum. The integration of Eq. (11.1) over the nucleus gives the relation

F(1−Av)Sr2

= 4σ(1−AIR)ST 4 + 4SZL. (11.7)

For reasonable values of the albedo Av and for the heliocentric distance less

than about 0.8 AU, the radiative contribution term in the above equation

can be neglected. Therefore Eq. (11.7) reduces to

(1−Av)S =4SZLr2

F=QLr2

F(11.8)

where Q = 4SZ represents the production rate of water which can be ob-

tained from the observed production rates of OH or H in comets (Chap. 6).


Nucleus 351

The value of L, the latent heat for vapourization of water is 11500 calo-

rie/molecule for T = 200K. From a knowledge of AS and (1 − Av)S it is

then possible to get the albedo and radius from the following equations:

S = (1−Av)S +AS (11.9)

and

Av =AvS

AvS + (1−Av)S.

The method is valid if the effective area is roughly the same for the

reflected light and for the vapourization equilibrium. However, there could

be a difference in the areas in a real situation. In addition, there is some

difficulty in separating the contributions to brightness from the nucleus and

coma. The quantity (1-A)S can also in principle be determined from the

infrared observations of the nuclei of comets. To a first approximation,

(1-A)S is directly proportional to the thermal emission in the 10–20 µm

region provided the vapourization process from the nucleus does not take

a significant fraction of the absorbed energy.

Even at large heliocentric distances a faint coma is still present. There-

fore direct photometric observations have the problem of separating the

light from the nucleus from the surrounding coma radiation. The coma

contribution can be minimized with the use of high spatial resolution ob-

servations of the nuclei of comets. In this approach the brightness of the

nucleus is observed as a delta function over a slowly varying coma contribu-

tion. Therefore the brightness distribution of a comet can be represented

as

B(ρ) = a[1/ρ+ bδ(ρ)] (11.10)

Where ρ is the projected distance from the nucleus. The first term in equa-

tion (11.10) represents the variation of the coma brightness for a constant

source with inverse square dependence of space density. The delta function

δ(ρ) represent the contribution arising from the nucleus, while a and b are

constants. A fit to the observed brightness variation will give the contribu-

tion from the nucleus which can be used to get the radius of the comet for

an assumed value for the albedo. If the rotational light curve is available it

is possible to get the semi axes for the nucleus.

The direct determination of the size of the nucleus of a comet was made

possible by the study of Comet Halley. The images taken by the spacecrafts



directly give the projected dimension. Based on different aspects seen by

the three spacecrafts, Vega 1, 2 and Giotto, it is then possible to reconstruct

the actual three dimensional shape of the nucleus. Figure 11.5 shows one

best image obtained from Giotto. It clearly shows the irregular shape of the

nucleus. Other comets which have been studied with spacecraft are Borrelly,

Wild 2 and Tempel 1 etc. The albedo can be estimated from a knowledge of

the area of the nucleus and the observed photometric magnitudes (equation

11.9).

The radius and albedo of cometary nuclei for comets are given in Ta-

bles 11.1 and 11.2. There is some variation in the observed radius, while

albedos have low values. Low value of albedo indicate that the cometary

nuclei are very dark.

Table 11.1 Sizes of cometarynuclei.

Comet a × b × c

(km × km × km)

Halley 7.65 × 3.61 × 3.61

Tempel 2 8.2 × 4.9 × 3.5Borrelly 4.0 × 1.6 × 1.6

Tempel 1 14.4 × 4.4 × 4.4

Wild 2 1.65 × 2 × 2.75

Table 11.2 Albedo and radius of cometary nuclei.

Comet Geometric albedo Radius(km)

Eucke 0.046 2.3

Tempel 1 0.05 3.1Borrelly 0.03 -

Neujmin 1 0.03 10.6Halley 0.04 -Hale-Bopp 0.04 30

Swift Tuttle 0.02 11.8IRAS-Araki-Alcock 0.03 3.5Arend-Rigaux 0.04 4.6

Lemy, P.L., Toth,I., Fernandez, Y.R. and Weaver, H.A.,2004, In Comets II, eds. M.C. Festou, H.U. Keller andH.A. Weaver, Univ. of Arizona Press, Tucson p. 223.


Nucleus 353

Fig. 11.5 Composite image of the nucleus of Comet Halley made from several images

obtained from Halley Multicolour Camera on board Giotto Spacecraft (Courtesy Keller,

H.U., Copyright Max-Planck - Institut fur Aeronomie, Lindau, Germany).

11.5. Mass, Density and Surface Gravity

Mass

It is rather difficult to make an estimate of the mass of the nuclei of



comets. A rough estimate can be made from a knowledge of their sizes and

densities.

Another method that has been generally used for getting the mass of the

nucleus is through change in the orbital period caused by non-gravitational

force arising from the sublimation of water ice from a rotating nucleus. This

force arising from the outflow of the material is related to the mass of the

nucleus of the comet through Newton’s second law of motion, F=ma. This

method has been applied to several comets. The derived masses lie in the

range of around 1 to 5 × 1013 kg. The derived mass for Comet Tempel 1

from the same method is (5.8 ± 3.0) × 1013 kg.

The study of the ejecta plume created by the impactor in Comet Tempel

1 by the Deep Impact mission has been used to make an estimate of the local

gravity at the impact site. This is used for making an estimate of the bulk

density for an assumed shape and a uniform mass distribution inside the

nucleus. The derived mass is 4.5 × 1013 kg. This value based on direct ob-

servations is consistent with the mass derived above from non-gravitational

force modelling. This therefore gives credibility to the indirect method

of estimating the mass of the nucleus of comets through non-gravitational

force modelling technique.

The perturbation produced by a close encounter of flyby spacecraft with

a comet could in principle be used to extract the nuclear mass. However

the expected perturbation is below the detectable limit of the present day

instruments.

The discovery of satellites around asteroids has raised the prospect of the

likely presence of satellites around cometary nuclei. The main advantage

of a binary system is that it is possible to derive the mass of the system

using Kepler’s laws from the observed orbits. However the detection of a

satellite companion to cometary nucleus is difficult because of its small size

and also due to the presence of coma even at large heliocentric distances.

If and when cometary binaries are detected, it will give a direct method of

estimating the mass of the nuclei of comets.

Density

The determination of bulk density of the nucleus of a comet is important

as it can give information about the structure of the nucleus. It can be

estimated provided the mass and the total volume (including voids and

pore spaces) of the nucleus are known.

Volume can be estimated from direct spacecraft imaging or from tele-

scopic observations that can yield the appropriate radius and axial ratios


Nucleus 355

of individual nuclei. Spacecraft imaging has been carried out for several

comets such as Halley, Tempel 1, Wild 2, Borrelly etc.

The charge-coupled device (CCD) photometry of nuclei when they are

far from the sun (and presumably inactive) provide an estimate of their

brightness. It is possible to make an estimate of the radius of the nucleus

and hence volume for a typical cometary albedo of 0.03 - 0.04.

The estimated bulk density lies in the region of around 0.2-0.8 gm/cm3.

The density of the fully compacted comet material consisting water-ice,

organics and dust is expected to be in the range 1.2 < ρcompact < 1.65

gm/cm3. Therefore the results appear to indicate a somewhat lower value

for the bulk density of the nucleus of comets. This indicates that the nucleus

could be fluffy. The estimated bulk poracity of the nucleus is about 0.3 to

0.6.

Surface Gravity

The impactor from the Deep Impact spacecraft that collided with Comet

Tempel 1 made a crater and ejected the material giving rise to an ejecta

plume. The ejecta plume was imaged continuously from the flyby space-

craft at a distance of around 700 km. These images show clearly that the

ejecta cone remained attached to the surface throughout the encounter.

This interesting observation indicates that the formation of the crater was

controlled by gravity.

The images of the ejecta plume taken by the spacecraft showed the

expansion of the base of the conical solid ejecta with time. The particles

forming the solid ejecta follow paths in the gravitational field of Comet

Tempel 1 at any given time. Making few assumptions such as, the cone is

nearly axially symmetric, scaling relations derived from laboratory studies

for gravity dominating cratering is valid and the properties of ejecta flow

could be extrapolated to this observed impact, it was possible to make an

estimate of the strength of gravity from the observed expansion of the base

of the conical ejecta. The estimated local gravity at the impact site of

Comet Tempel 1 is about gnuc =3.4 × 10−4 m/sec2.

11.6. Rotation

The rotational state of the nucleus of a comet is important for an un-

derstanding of the physical properties of the nucleus of a comet and its

surface. The evolution of the spin state and the nucleus gives information



about the changing pattern of the observed activities of a comet. It also

plays a key role in a better understanding of the activity of the nucleus

of a comet resulting due to gas evaporation from the nucleus, generally

called ‘non-gravitational force’. These studies are being carried out obser-

vationally to deduce the rotation period of the nucleus and the orientation

of the spin axis of comets and its interpretation in terms of physical and

dynamical models.

The time period of nuclear rotation can be estimated from some time-

dependent property associated with the comet. It could be the observation

of a certain repetitive feature, sequence of images of the nucleus, from an

analysis of modelling of light curves or the non-gravitational force and so

on.

The successive expanding halos with velocities of about 0.5 km/sec seen

in the photographic observations of several comets have been interpreted

on the basis of the rotation model for the nucleus. Parabolic envelopes

have also been seen in the coma of comets. With the assumption that these

arise out of the repetitive ejection of material from a single active area, it

is possible to make an estimate for the rotation period of the nucleus. The

procedure is to get the time at which the material was ejected, called the

zero time. If the zero time can be found from concentric halos, then the

time interval between the two zero times gives the rotation period. From

the measured diameters of the halos or from the latus recta of the parabolic

envelopes, the rotational velocities of a large number of comets have been

determined. This method is generally known as ‘zero age’ method. The

basic assumption involved in the halo method is that the halo structure

arises mainly due to the rotation of the nucleus.

Many comets seem to show broad, fan-shaped coma coming out of the

central condensation and in the general direction of the Sun. The width

appears to change from comet to comet and with time. These have been

interpreted in terms of anisotropy in the out-gassing. This can clearly be

seen from the densitometer tracings of the inner part of the Comet Pons-

Winnecke which are shown in Fig. 11.6. These have been interpreted in

terms of the orientation of the spin axis of the nucleus.

The continuous monitoring of continuum or molecular emission lines

over an extended period of time can give information about the periodici-

ties connected with the rotation of the nucleus. The rotational light curves

derived from photometric observations when the comet is far off from the

sun can be used to extract the shape and the rotational period of the nu-

clei of comets. The idea being that the periodic temporal variation of the


Nucleus 357

Fig. 11.6 The densitometer tracings of the photograph of June 25, 1927 in the inner

part of the fan of Comet Pons-Winnecke are shown here. The jet near the nucleus in the

sunward direction and the curvature effect as one goes outwards can be seen. The circleddot denotes the direction of the Sun. (Sekanina, Z. 1981. Ann. Rev. Earth Planet Sci.,

9, 113). Reproduced with permission from Annual Reviews Inc.

brightness is interpreted in terms of the rotation of an elongated body.

At present observations are carried out at visible and infrared wavelength

regions for producing light curves. High spatial resolution observations

carried out with HST at various times can be used to derive the flux con-

tribution from the nucleus and hence the rotational light curve. The time

period can be extracted from the light curve through various methods such

as Autocorrelation analysis, period dispersion minimization method etc.

The derived rotational time period for Comet Tempel 1 is 41.27 h. Based

on various methods, the time periods of rotation of many comets have been

determined. Unfortunately there is lot of variation in the derived periods

for the same comet. The diversity in the derived values are illustrated with

two examples.



For Comet Encke, the zero age method gave a value of 6.5 hrs. The pe-

riodic variation in the photometric magnitude gave a value of 22.4 hrs. The

photometric data obtained near aphelion could be satisfied with periods of

22.6 hrs, 15.1 hrs,11.1 hrs or 7.5 hrs. Later studies have put some doubt

on 15.1 hrs period. It is possible that the observed periods may represent

harmonically related periods. If this is the case, then the observed peri-

odicities could relate to the basic period of 7.4 h. However it is not clear

whether this is the case.

For Comet Halley several values have been derived. The ‘zero age’

method gave a period of about 10 hrs. The most direct evidence came from

the morphology of sun ward jets. A rotational period of around 52 hours

was estimated from the processing of spiral jets seen in the photographs

taken during the 1910 apparition. The comparison of Vega 1 and Vega 2

camera observations gave a rotational period of 52–54 hours. However the

images of features like jets near-nucleus show a period ≈ 2.2 days. A time

period of about 7.4 days was obtained from the time series studies of the

narrow-band photometric observations of the coma. The same time period

was also indicated by the morphology of jets of free radicals, Lyman α

emission and the 18 cm OH line, but nothing around a period of 2.2 days.

Therefore there is conflicting results derived from various methods. Images

taken from Vega 1, Vega 2 and Giotto spacecrafts could refer to different

viewing geometries and timings of the nucleus of Comet Halley that might

complicate the interpretation of the observed images. But on the whole

there appears to be two rotational periods corresponding to 2.2 and 7.4

days.

For a non-spherical nucleus there could be several possible rotation

states. It can rotate around the long or short axis. It can also have nu-

tation motion. The rotation could be in its lowest rotational state energy

or in an excited state. Depending on the state of the nucleus, it can have

one or two time periods and their harmonics. Several attempts have been

made to understand the observed periodicities of Comet Halley in terms

of rotational, precessional or nutational motion of the nucleus. From a de-

tailed model study of the nucleus of Comet Halley, it has been possible to

understand the observed time periods. In this model the nucleus is con-

sidered to be a prolate spheroid with dimensions of 17 × 8.5 × 8.5 km.

It takes into account five active areas on the surface of the nucleus that

are active only when they are exposed to the sun and provide torques to

the nucleus. The model can explain reasonably well the ground based data

as well as the interpretation of visible data from spacecraft. The resulting


Nucleus 359

model indicate that long axis of the nucleus precess around the rotational

angular momentum vector in 3.69d at an angle of 66. In addition the long

axis rotates around its own axis with a period of 7.1d.

A rotation period of 26 h has been deduced for Comet Borrelly based

on ground based and HST observations. A linear jet emanating from the

nucleus was seen from the Deep Space 1. Several observations about this

feature indicate that the jet is along the rotation axis of the nucleus and

also spinning close to its lowest energy spin state.

Based on the above brief discussion, it is evident that the periodicities

depend upon the state of the nuclei. In particular, on the shape, activity

and their location, jets, how the gas comes out and so on. This leads to the

complex nature of the periodicities in the observed spectra than indicated

by the assumed simple models. Hence the interpretation of the observed

periodicities is a difficult matter and is comet dependent.

11.7. Nucleus Composition

The major goal of cometary studies is to determine the chemical com-

position of the nuclei of comets. Therefore the evolution of the activity of a

comet along its orbit is important for an understanding of the chemical na-

ture of the nucleus. In view of this, extensive studies have been carried out

with regard to the variation of gas production ratio of species as a function

of the heliocentric distance for various comets. The observed production

rates of various species are found to vary with heliocentric distance and

from comet to comet. Therefore the coma abundances of species cannot

directly reflect the composition of the nucleus of a comet. This problem

has been investigated based on the model for the nucleus.

As discussed earlier the structure of the nucleus of a comet is a com-

plicated function of various physical processes that could take place inside

the rotating porous nucleus containing amorphous ice and volatiles. These

result in structured layers with different composition. The top layers con-

taining the least volatile material and the deepest layers containing the

most volatile species. This structured layer nucleus can be used to calcu-

late the production rate of the species and the integration over the entire

orbit will give the total amount of material ejected during one orbit of the

comet. The deduced abundance ratios for each specie can then the com-

pared with the abundance ratios for the same specie in the nucleus. The

results of such studies have shown that the abundance ratios of the ob-



served species in the coma integrated over the entire orbit of the comet is

a better representation of the composition of the nucleus of a comet. This

implies that one has to take into account the change in coma composition

with heliocentric distance.

Comet Hale-Bopp was a bright and active comet which made it possible

to study this problem in greater detail. The comet was observed over a large

range of heliocentric distances of 1 to 7 AU. The expected abundances in

the coma from the structured layer chemical model that takes into account

the gases arising from, the surface of the nucleus (sublimation), the interior

of the porous nuclei and the distributed source, can be calculated. These

results can be compared with the observed ratios to deduce the composition

of the nucleus. A mixture composed of 35% amorphous H2O, 7%CO2,

13%CO (50% trapped in the amorphous ice) and 45% dust give a good fit

to the observed data as can be seen from Fig. 11.7. The fit to the observed

data will be much better if the distributed source of CO were subtracted.

The deduced abundance ratio for the nucleus is similar to the expected

abundance ratios from a condensable component of molecules at a lower

temperature from a mixture of solar abundance elements.

11.8. Mass Loss

The volatiles sublimating from the nucleus of a comet, eventually leave

the system. Therefore the mass loss of volatiles in comets can be used to

make an estimate of the lifetime of a comet. For this purpose, the observed

mass loss rate of water from the nucleus of a comet per revolution may be

used. For Comet Borrelly, the total amount of water sublimated from the

nucleus during an apparition is around 1.1 × 1010 kg and the estimated

mass of the nucleus from non-gravitational force modelling is around 1 to

5 × 1013 kg. Therefore the comet can last for about 1000 to 5000 passes.

Taking a typical value of 5 yrs for the time period for short period comets,

the observed mass loss rates indicate that the lifetime of a short period

comet is limited to about 5 × 103 to 3 × 104 yrs.

When most of the volatiles are lost from the nucleus, it might go into

an extinct state. It is possible that some of them may be in the asteroidal

belt. As more and more volatiles sublime, the very low tensile strength

material of the nucleus may become more fragile leading to disintegration

of the nucleus into large number of pieces, which then takes on their own

path.


Nucleus 361

Fig. 11.7 Mixing ratio Q(CO)/Q(H2O) is plotted as a function of the heliocentric dis-tance for Comet Hale-Bopp. The dashed line is the model fit. The model fit will be much

better if the distributed source is subtracted from the observations (Huebner, W.F. and

Benkhoff, J. 1999. Space Sci. Rev., 90, 117).

11.9. Structure

There is no direct method of knowing the internal structure of the nu-

cleus. Therefore one has to infer the possible nature of the structure of the

nucleus from the observed characteristics of comets. The observational fact

that the splitting of the nuclei is quite commonly seen in comets implies

that the nuclei should be quite brittle. Otherwise it will not break that

easily. The important point is that the splitting process does not totally

destroy the nucleus. Comets split into several components when it passes

within the Roche limit of the planet. The Roche limit is defined as the dis-

tance from the planet at which a smaller body like a comet which is porous



and with weak tensile strength will be torn apart due to tidal forces. The

classic example is the Comet Shoemaker-Levy 9 which came with in 0.001

AU of Jupiter in 1992 and disintegrated into 21 pieces due to tidal forces

(Figs. 1.8, 1.9). The splitting phenomenon also gives a chance to compare

the different split components with the original nucleus and through this

one may be able to get some information about the changes with depth.

The splitting of Comet West into four components (Fig. 1.7) has been an-

alyzed in some detail. The components are observed to separate out with

extremely small velocities of the order of one meter per second. After their

initial velocity they are all subjected to non-gravitational separation forces

produced as a result of the jet action resulting out of vapourization of ice

(Sec. 11.10). The observations can be fitted very well with such a phys-

ical model. Comet LINEAR (1999S4) split up into several pieces during

its passage around the sun in 2000. The observations of several other split

comets, which have different time-period can also be explained in a similar

manner. These results show that the different pieces coming out of comets

of extreme ages show the same behaviour with regard to the vapourization

process. The splitting of the nuclei of comets appear not to depend upon

whether the comet is of short period or long period or on their orbits or

happens before or after perihelion etc. The splitting of comets also ap-

pears to be more or less random in character. The outburst seen in many

comets indicate the nucleus could be fragile as well as porous. In general

the new comets appear to exhibit more activity than short-period comets

at large heliocentric distances. This may indicate that the outer layers of

new comets may be extremely loose with more volatile material contained

in it or there could be a halo of volatile material around the nucleus.

The erosion of the nuclear surface can take place during its long exposure

at far off distances from the Sun due to the bombardment of cosmic rays,

the solar wind etc.

The gross material strength of comets can be obtained from the esti-

mates of the tidal forces acting on the Sun grazing comets, split comets

and from other considerations. The gravitational compression force (<∼ 105

dynes/cm2) is quite small and cannot compress the icy material of the

nucleus. The tensile strength against the tidal disruption of Sun grazing

comets indicates a value in the range of 103 to 105 dynes/cm2. Similar or-

der of strength of the material in the nucleus is estimated from the period

of rotation of the nucleus. These values are small and they indicate the

material to be weak. The strength of the material can also be estimated

based on the study of meteor trajectories as they plunge into the Earth’s


Nucleus 363

atmosphere since they are believed to be cometary debris (Chap. 13). They

also give value of about 103 to 105 dynes/cm2. The estimated values for

the tensile strength for cometary material in the nucleus may be compared

with the values for solid water-ice and rocky material which is ≈ 4 × 106

dynes/cm2. It shows the cometary material to be of low material strength.

In addition to this, the particles collected at high altitudes appear to be

fluffy. The estimated low bulk density for the nuclei of comets indicate the

material to be fulffy and loosely packed in the nucleus.

There are also other types of observations which can give some clue to

the structure of the nucleus. It is well known that comets go around in

their orbits many times. These observations show that only a thin layer

of the material comes out of the nucleus keeping the core of the nucleus

intact. More support for this comes from the Sun grazing comets which

survive even after such an encounter. Also the jet force arising out of the

vapourization of the gases modifies only the trajectory of the comet and it

does not destroy the nucleus as a whole.

All these observations appear to show that the nucleus cannot be com-

posed of a cluster of small particles termed as the sand-bank model. In

the sand-bank model the particles are independent of each other but they

follow similar orbits around the Sun. In this case there is no real body at

the centre although there is a high concentration of particles at the cen-

tre. The images of the nucleus of Comet Halley, Tempel 1, Wild 2 and

Borrelly taken from spacecrafts have clearly shown the nucleus to be a one

solid chunk. The images have also given enormous information about the

surface morphology and topology of the nucleus of comets.

Modelling cometary nuclei which can reproduce the observed character-

istic behaviour of comets can also provide internal properties of a comet.

The earlier model based on the assumption that the solar radiation is the

only source of energy responsible for the observed cometary behaviour has

been superseded by more sophisticated models. This is due to the fact that

the cometary nucleus likely to contain amorphous ice and its subsequent

crystallization could provide an additional source of energy. In addition,

there could be another source of energy arising from short-lived radionu-

cleide 26Al. Models which take into account these three sources of energy,

namely solar radiation, through crystallization process and radioactive de-

cay and the high poracity of the nucleus have been considered. These

studies indicate the stratified structure of the nucleus (Fig. 11.3).

Comets are formed as a natural product in a contracting solar nebula.

The resulting nuclear structure could be aggregates of smaller icy planetes-



imals. This concept has led to models such as fluffy aggregaate, primordial

rubble pile and icy-glue. The collisional evolution appears to have played

a role as well for the cometary nuclei. Infact all the evidences seems to

indicate that the structure of nuclei of comets is processed rubble pile of

smaller ice planetesimals of previous generation. Therefore the general feel-

ing at the present is that the processed models such as fluffy aggregates or

the collisionally evolved rubble pile models are the best representation of

the interior structure of a cometary nucleus.

Fig. 11.8 Schematic diagram showing the persisting layers that have been seen on the

surface and deeper layers of Comet Tempel 1 based on the images taken by Deep Impactspacecraft. Such features has also been seen in Comets Wild 2 and Borrelly (Belton,

M.J.S. et al. 2007. Icarus, 187, 332).

The highest spatial resolution images of Comet Tempel 1 obtained from

Deep Impact showed clear evidence for the presence of layering over the

surface of the comet (Fig. 11.8). Such layering structures are also present

in the earlier images of Comet Borrelly and Wild 2. However the layering

present in Comet Tempel 1 is very conspicuous. The appearance of such

a feature on a large scale in three comets belonging to Jupiter family of

comets has led to the suggestion that layering could be a common feature

in the internal structure of these comets. These observations have been


Nucleus 365

interpreted in terms of a new proposed model called ‘talps’ or ‘layered pile’

model. In this model the interior of a cometary nuclei consists of a core,

overwhich a randomly staked layers are piled up (Fig. 11.8). This model

can explain several observed characteristics of comets. The consequence of

such a model is that the nuclei of Jupiter family of comets are primordial

in nature and the structure of the nucleus has not changed much from the

time of their formation.

11.10. Non-gravitational Forces

The orbit of a comet can be computed by well-known methods based on

Newton’s law of gravitation. The accuracy of the determined orbit depends

upon the number of observations available for the comet. The orbit of short-

period comets can be determined accurately as they can be observed many

more times than that of long-period comets. The orbit of a comet is gen-

erally perturbed strongly by the planets as they enter the inner part of the

solar system. The perturbation induced due to such an encounter can be

incorporated in the orbit calculation and these complications have become

easier to handle with the availability of high speed computers. However

in spite of such accurate orbit calculations, it has been noticed that they

seem to differ from the observed position of comets at each revolution. The

well-known comet for which this is striking is the Comet Encke, discovered

in 1786 and having a period of 3.3. years. It persists in returning at each

revolution about 2 12 hours too soon. This was noticed by Encke himself as

far back as 1820. The deviations from the observed position of comets are

typically a fraction of a day and is more in some comets. Observationally

it is found that out of several comets studied some came earlier than the

predicted time and others later than the predicted time. This means that

some are accelerated and the others decelerated, which demonstrated con-

clusively that there was an additional nongravitational force acting on the

comet which was not included in the orbit calculations. It was noticed from

observations that this nongravitational force decreased substantially with

an increase in the distance between the comet and the Sun. Enckes first

suggestion that the observed deviation could arise from a resisting medium

was not in conformity with the observations. However the basic model

based on the sublimation of gases from the icy conglomerate nucleus has

been able to explain the observations reasonably well. Refinements have

been considered to this model due to the presence of complex features on



the surface of the nuclei of comets.

The equation of motion of a comet in the rectangular coordinate system

used in the orbit calculation procedure can be written as

d2r

dt2= −µ r

r3+∂R

∂r(11.11)

Here µ is the product of the gravitational constant and the mass of the Sun.

The term on the right-hand side of Eq. (11.11) takes care of the planetary

perturbations i.e., R is a planetary disturbing function. The above equation

was generalized to include nongravitational forces as

d2r

dt2= −µ r

r3+∂R

∂r+ F1r + F2T + F3N. (11.12)

Here F1, F2 and F3 represent the additional acceleration components, F1

is along the radius vector defined outward along the Sun-Comet-line, F2 is

perpendicular to r in the orbit plane and towards the comet’s direction of

motion and F3 normal to the orbit plane. r, T and N are the three unit

vectors along the three direction of forces. Since the nature of F1, F2 and

F3 was not known, one can write

Fi = Aif(r) (11.13)

and to see what form of f(r) can remove the observed discrepancy. The

component of the force F3 is generally present for active comets, but it

is difficult to determine a meaningful solution due to its periodic nature

and also the average non-gravitational acceleration is determined from the

solution over three or more apparitions. Therefore for most of the comets

the value of A3 is put equal to zero. The change in the semi major axis in-

troduced due to the radial and transverse perturbing acceleration (RP , TP )

is given by the equation

da

dt=

2

n(1− e2)1/2

[(e sin ν)RP +

P

rTP

]where n, e, ν and r represent the orbital mean motion, eccentricity, true

anomaly and the heliocentri distance respectively. P presents the orbital

semi-latus rectum, a(1−e2). The derived empirical function f(r) was found

to agree well with the form g(r) given by the Eq. (11.5). Knowing the

function f(r), the constants Ai can be calculated. These nongravitational

parameters are actually obtained from the least square fit to the definitive

orbit of the comet. Therefore the empirical work on the nongravitational

force has been put on a firm basis based on a physical model. In recent

years the expression (11.5) has been used in all the orbit calculations.


Nucleus 367

These ideas fit in well with the matter ejected from a rotating icy model

of the nucleus which can be understood from Fig. 11.9. When the solar

radiation impinges on the nucleus, the material evaporates from the surface

and moves out in the direction of the incident radiation. This results in a

jet action, a force pushing the comet away from the Sun. If the nucleus

is rotating then there will be a time delay between the heating and the

ejection of the gas from any point on the surface. If the rotation is in the

same direction as its motion around the Sun, the delayed jet action will

have a forward component which will increase the orbital period of the com

et (Fig. 11.9(b)). As a result it will go into a higher orbit and so it will show

up later than the predicted time. On the other hand, if the nucleus rotates

in the opposite direction to the motion around the Sun (Fig. 11.9(c)) the

jet action will have a backward component. This will reduce the period

of the comet and hence it will show up earlier than the predicted time.

Therefore, depending on the direction of rotation of the nucleus, some of

them will be accelerated and some will be retarded, which is consistent with

the observation.

Fig. 11.9 The result of outguessing on the cometary nucleus: (a) the nucleus does notrotate; (b) the nucleus rotates along the direction of motion around the Sun. The jet

force is also along the direction of motion increasing thereby the period of the comet;

(c) the nucleus rotates in the opposite direction to the direction of motion around theSun. The jet force is now in the opposite direction to its motion resulting in a decreasein the period of the comet.

It is now possible to make a meaningful comparison of the nongravi-

tational parameters A1 and A2 for different comets as they represent the

relative mass loss rates. However, there was still a major problem requir-

ing the outguessing to be much higher than what was known at that time

based on the line emission of C2 and CN in the visual region of the spectrum

(Chap. 6). Otherwise the jet force was not enough to exert an adequate

force on the comet. This problem was resolved with the discovery of the



presence of a huge halo of hydrogen gas around Comet Bennett in the 1970’s

through the detection of the Lyman α line radiation (Chap. 6). The ob-

served decrease of nongravitational effect with an increasing distance from

the Sun can also be explained based on the vapourization model. The

nongravitational parameter indicates that about 1% of the material is lost

from each comet in one revolution which is consistent with other estimates

(Sec. 11.8). To take care of the active emission areas from the surface of

the nucleus, an effective sublimation rate Z(z, r) (molecules/cm2/s) at a

heliocentric distance r and the Sun’s local zenith angle z is defined as

Z(z, r) = Z0(r)G(z, r)

where Z0 is the sublimation rate at the subsolar point and G(z, r) (≤ 1) is

the relative sublimation rate at the Sun’s Zenith angle z. If there are several

sources then it has to be averaged to get the rotation-averaged acceleration.

The transverse component given by the parameter A2 is quite sensitive

to the effects of episodial events in the orbital motions of comets. The

determination of non-gravitational parameters require at least three con-

secutive apparition and in addition at least three independent values of the

parameters A2 is required for the study of temporal variations. This limits

the study to short period comets for which astrometric observations have

been made for 5 or more apparitions. Such observations have been used to

determine the values of A1 and A2 for a large number of comets. These have

been used to understand the diurnal effects and seasonal effects in comets.

Diurnal effect refers to the short term variations arising due to outgassing

from uneven exposure of individual sources to sunlight caused by the rota-

tion of the nuclei. Seasonal effects are the long term variations associated

with producing the asymmetric curve in the production rate with respect

to perihelion.

The non-gravitational parameter for the Comet Kopff which was posi-

tive before 1930 changed over to negative value around 1940. On the other

hand for Comet Halley the value of A2 has not changed over a long time

period. This indicate that Comet Halley has been out-gassing at almost

the same rate over this long period of time.

In the discussion so far, it has been generally assumed that the parame-

ter representing the component of the force normal to the orbital plane, A3

is equal to zero. This is due to fact that the introduction of the parameter

A3 did not seem to improve the orbital solutions implying that it is hard

to extract the value of A3 i.e. essentially indeterminate. Therefore A3 is

the least understood parameter and it requires further study. With this


Nucleus 369

in view, several attempts have been carried out, in recent years, to detect

the contribution from the normal component of the transferred momen-

tum. Using various mathematical techniques, the three non-gravitational

parameters A1, A2 and A3 have been derived for a number of comets. The

results are rather consistent in showing that the value of A3 is not large.

Therefore with the availability of accurate values for the parameters A1,

A2 and A3 representing the component of non gravitational perturbation

in the radial, transverse and normal directions it may be possible to study

the time evolution of the orbits, precession of the spin axis, wobbling and

so on.

All the studies carried out so far for the last two decades or so, were

based on the symmetric models for the non-gravitational effects in the equa-

tion of cometary motion. The implicit assumption in the symmetric model

is that the non-gravitational effects are the same on either side of the per-

ihelion position. However, in a real situation the non-gravitational effects

are more of perihelion asymmetrical in nature. The evidence for such an

effect comes from the non-random distribution of the active regions on the

surface of the nucleus and the observed asymmetrical nature of the light

curve of comets, which basically reflect their out gassing histories. There-

fore the results based on symmetrical non-gravitational acceleration model

is being questioned. However, it should be noted that the symmetric mod-

els have been very successful in providing accurate ephemerides but the

derived physical properties of comets should be taken with caution. There-

fore attempts are being made to take into account the asymmetrical nature

of the non-gravitational force in comets by considering the out gassing as

accurately as possible.

An approach that has been attempted is a slight modification of the

symmetrical non-gravitational acceleration model. In this procedure the

possible asymmetry in the out-gassing is taken into account by shifting the

time (DT) for a few days before or after perihelion passage so that it co-

incides with the maximum value of the water vapourization curve. DT=0

corresponds to the symmetric case. Therefore the expression for the asym-

metric non-gravitational acceleration model at any time t is given by g(r′),

where r′ = r(t′) = r(t − DT ). The new calculations are similar to the

symmetric model case except that the function g(r′) is evaluated for the

heliocentric distance corresponding to the time t − DT , rather than for t

itself. The time shift DT is varied till a best fit to the anisotropic observa-

tions is found and this in general agrees with the shift seen in the maximum

of the light curve from the perihelion. It is found that the asymmetric



model appears to provide better orbital solution compared to symmetric

models. The new approach is being applied to the study of several comets.

The results based on a few comets have shown that the radial and trans-

verse non-gravitational parameters derived from non-symmetric models are

different from symmetric models and hence in their inferred physical prop-

erties. Over long periods of time the value of DT may change its value

and sign. The value of DT is about 40d for Comet d’Arrest and is also

stable over many observed apparitions of the comet. On the other hand for

Comet Giacobini-Zinner DT = + 14 days during 1959–1973 and - 13 days

during 1972–1987. For comet Halley DT = + 20 days and this is consistent

with the post-perihelion peak observed in the light curve. There are some

comets which show long term non-gravitational effect to be irregular as well

as the value change rapidly. These could be understood in terms of forced

precession model of a rotating non-spherical cometary nucleus.

Since the non-gravitational force is the resultant effect of the redistribu-

tion of the momentum that is transferred to the nucleus by the sublimating

gas, it is a complicated function of several factors like shape, structure,

nature and areas of active regions, jets, direction of the ejection of gas and

so on of the nucleus. The resultant effect is to complicate the rotational

as well as the orbital motion of the nucleus of a comet, which shows up as

peculiarities in the observations (Fig. 11.10). It also depends upon sudden

activation or deactivation of discrete emission centres in the nuclear surface.

All these complicated effects which varies from comet to comet and

affect the non-gravitational perturbations have to be considered for getting

accurate orbit of comets.

11.11. Ortho to Para Ratio of Molecules

The nuclear spin of two similar atoms in a molecule such as H2, give rise

to segregation of rotational levels of the molecule depending upon the total

nuclear spin angular momentum (I). Therefore it was postulated based on

Quantum Theory the existence of two forms of molecular hydrogen called

Ortho - H2 and Para -H2 and got experimental confirmation of this predic-

tion, a great success story. Later work found that in ortho - H2, the two

hydrogen atoms spin in the same direction (I=1,↑↑) and have rotational

levels with odd J values. In para - H2, the nuclei of two hydrogen atoms

spin in the opposite direction (I=0,↑↓) and have rotational levels with even


Nucleus 371

Fig. 11.10 A composite figure showing the light curves of Comet Schwassmann-

Wachmann 2 for the period from 1929–1987. The negative times correspond to theinbound orbit and positive times to outgoing orbit. The long-term variations as well as

erratic behaviour can be seen (Sekanina, Z. 1993. Astron. Astrophys. 271, 630).

J values. Radiative and collision transition betwen the rotational states of

ortho and para molecules are strongly forbidden.

The H2O molecule also exists as ortho - H2O(I=1) and para - H2O(I=0)

depending upon whether hydrogen nuclear spins are in parallel or in anti-

parallel state with statistical weight of (2I+1). The lowest rotational level

of ortho - H2O is higher than the lowest level of para - H2O by 24 cm−1.

Therefore the ratio between the total populations of ortho - H2O and para

- H2O states depend upon the temperature called the Spin temperature.

This temperature refers to the temperature for a given ortho to para ratio

under local thermodynamic equilibrium.

In thermal equilibrium the ortho-to-para ratio (OPR) can be determined

from the rotational distribution of the molecules

i.e., OPR =(2Io + 1)

∑(2J + 1) exp

[−EokT

](2Ip + 1)

∑(2J + 1) exp

[−EpkT

] .

Here o and p refer to ortho and para rotational levels. J and E refer

to rotational quantum number and energy of the levels respectively. The

above equation can be evaluated as a function of temperature using energy



level diagram of H2O molecule. The results are shown in Fig. 11.11. For

high temperatures the population ratio of ortho-to para H2O approach the

statistical weight 2I+1 which is 3 for H2O. As can be seen from the figure,

the statistical equilibrium value of OPR=3.0 is achieved for temperatures

greater than around 60K. For lower temperatures the OPR is less than the

statistical equilibrium value.

Fig. 11.11 The time dependent Ortho to Para ratio is shown by the solid curve. The

observed ratio for H2O in various comets is also shown (Bonev, B.P. et al. 2007. Ap.

J., 661, L97).

Since the nuclear spin conversion in H2O is forbidden during radia-

tive transitions or non-destructive collisions, the OPR formed at a partic-

ular temperature is essentially frozen-in and OPR is believed to remain

the same throughout the coma. Therefore the observed OPR should give

an indication of the spin temperature that existed when water molecule

was formed and hence the location in the solar system where comets were

formed. Therefore ortho-to-para abundance ratio of cometary H2O could

give information about the region of formation of comets in the solar sys-

tem about 4.6 Gyr ago. In view of this, the determination of ortho-to-para

ratio of H2O in comets is of particular interest.

The first determination of ortho-to-para ratio of H2O was for Comet


Nucleus 373

Halley in 1985-86 based on the observed lines of ortho and para in the

2.9 µm spectral region. Since then OPR ratio of water has been determined

for a large number of comets. This is determined from a comparison of the

observed spectra with the synthetic spectra based on fluorescence model.

The molecules NH3 and CH4 which exhibit ortho and para states and

whose lines lie in the radio and infrared spectral regions have also been

used to determine the OPR in comets. Although OPR ratio of NH3 can be

determined directly from their observed spectral lines, it is more advanta-

geous to get this ratio from the study of NH2 molecule whose lines are quite

strong and lie in the visible spectral region. As a typical case, a comparison

betwen the observed spectra and the best fit synthetic spectra for NH2 is

shown in Fig. 11.12 for Comet Ikeya-Zhang.

Fig. 11.12 Comparison between the observed and modeled spectra of NH2 A (0, 9, 0) -X (0, 0, 0) band in Comet Ikeya-Zhang for April 19, 2002. The best fit for OPR of NH2

is 3.22 for the model spectrum indicating spin temperature of 32 K for NH3 (Kawakita,H. et al. 2004. Ap.J., 601, 1152, Reproduced by permission of the AAS).

The observed OPR of NH2 is used to get the OPR of NH3 with the

assumption that NH2 is the major photodissociation product of NH3 and

the application of nuclear spin selection rule to photodissociation reaction

of NH3 into NH2 and H is valid. The spin temperature determined from

the study of H2O, NH3 and CH4 for several comets are given in Table 11.3

and shown in Fig. 11.11 for H2O. The striking feature of Table 11.3 is the



grouping of the observed spin temperature of H2O, NH3 and CH4 around

30 K with some scatter, even though the observations refer to different

heliocentric distances, gas production rates or the orbital period of comets.

This indicate that the spin temperature appears to reflect the temperature

of the region of formation of these molecules and hence of comets. If the

spin temperature reflect the place of formation of comets in the solar nebula,

then one expects spin temperature of Jupiter family of short period comets

(originally in the Kuiper Belt region which is farther than 30 AU in the

solar nebula) must be smaller than Oort cloud comets (originally in the

region of giant planet region of 5–30 AU from the sun). However the spin

temperature of these two types of comets indicate that both Oort cloud

comets and Jupiter family comets were formed under similar conditions

in the solar nebula and hence have a common origin. This is possible if

the cometary material had originated from a presolar molecular cloud at

30 K. Even though the observed spin temperature of H2O, NH3 and CH4

of comets cluster around 30 K, there are deviations as well. Of particular

interest is Comet Wilson and other comets whose OPR of H2O is close to the

statistical equilibrium value of 3(spin temperature is >∼ 50 K). These have

raised the question of the real meaning of the observed OPR of molecules

in comets and the corresponding spin temperature.

Although the nuclear spin conversion is forbidden, the constancy of

nuclear spin temperature over large time scales is not clear at the present

time. Some laboratory studies have indicated that the radiative nuclear

conversion time for H2O is much larger compared to that of CH4. Also if

equilibrium took place within the coma, the spin temperature should vary

with heliocentric distance of the comet. On the other hand if equilibrium

took place inside the nucleus, the nuclear spin temperature should vary with

the orbit of the comet. i.e. the spin temperature should differ between short

period and long period comets. Therefore laboratory study of cometary

analogs can help in a better understanding of the conditions under which

nuclear conversion can take place. This in turn can help in clarifying to

what extent the observed spin temperature represents the formation history

of comets.


Nucleus 375

Table 11.3 Nuclear spin temperature in comets.

Comet H2O NH3 CH4 Orbit

Halley 29±2 OCWilson > 50 OC

Hale-Bopp 28±2 26+10−4 OC

Hartley 2 36±3 JF

Lee 30+15−6 OC

S4 LINEAR ≥30 27+3−2 OC

A2 LINEAR 23+4−3 25+1

−2 OC

WM1 LINEAR 30+5−3 OC

Ikeya-Zhang 32+5−4 OC

NEAT 31+11−5 31+4

−2 33+2−1 OC

MACHHOLZ >34 OCTempel 1 24± 2 JF

Enake ≥33 JF

Kawakita, H. et al. 2006, Ap.J., 643, 1337. Bonev,

B.P., Mumma, M.J., Villanueva, L. et al. 2007,Ap.J., 661, L97. Kawakita, H. et al. Icarus 2007,

127, 272. OC (Oort cloud); JF (Jupiter family)

11.12. Binary Systems

The discovery of binaries among Asteroids, Near Earth Asteroids,

Transneptunian objects and Trojan Asteroids have raised high prospects

of finding satellite around cometary nuclei. The detection of a satellite

around the cometary nucleus will be of great significance as it will provide

a direct method of getting the mass of the cometary nucleus. If the mass is

known and size can be determined by an independent method, the density

of the material can be determined. This will give an insight for an under-

standing of the properties of the nucleus such as, poracity, structure of the

nucleus etc. However in a real situation, detection of a binary is a difficult

task as the nucleus itself is small and in addition coma is present even at

large distances. Hence, so far there is no definite observational evidence for

the existence of satellites around cometary nuclei.

There are several possible scenarios for the formation of a satellite

around a cometary nuclei. Capture of an external object is highly un-

likely process. The nucleus can fragment and the pieces can assemble to

reform into a satellite. It could also bi-furcate into two co-orbiting bodies.

The splitting of cometary nuclei is a common phenomena. So the frag-

ments can be produced. However due to non-gravitational force, the frag-

ments may drift apart quite fast and not allowing for the formation of a



stable binary configuration. Even if a satellite can form, its motion around

a non-spherical rotating and an active cometary nucleus could be highly

complex. Therefore the detection of a satellite around a cometary nuclei is

a real challenge from the observational point of view.

11.13. Summary

The vapourization theory of water from the nucleus could explain rea-

sonably well the visibility of comets at around the heliocentric distances

of 3 AU. However, it had difficulty in explaining comets seen at distances

much beyond 3 AU. This led to clathrate model. Such a configuration is not

practicable as it can be produced only at high pressure conditions, which

is not the case in the case of comets. The working model at the present

time is that the nucleus contains amorphous ice, and release energy when

it crystalizes. Consequently, the repetition of heating and cooling processes

inside the nucleus creates a layered structure with different compositions.

The most volatiles lie in the near central regions and less volatiles near

the surface layers of the nucleus. This model can reasonably explain all

the observed activities of a comet such as bursts, random or repetitive,

outbursts at large heliocentric distances etc. The flybys to several comets

have given surprising results with regard to the morphological structure of

the nucleus. The tensile strength of the nucleus is quite low and hence it

is fragile. Therefore the nucleus can break up easily, which has been seen

in several comets. The ortho-to-para ratio of several molecules indicate a

formation temperature ∼ 30 K which implies that comets are formed in the

low temperature region of the solar nebula. The surface gravity of Comet

Tempel 1 has been determined for the first time from the observation of

ejecta plume. The non-gravitational force which arise due to the net trans-

fer of momentum to the nucleus, is a complicated function which depends

upon the shape, active areas and their location, rotation, jets, outbursts

etc. of the nucleus. Similarly all these effects has influence on the nuclear

rotation giving rise to one or several time periods for the rotation of the

nucleus of a comet.


Nucleus 377

Problems

1. Do you expect the vapourization rate of gases from the nucleus to be

constant at every revolution around the Sun?

2. Consider a nucleus made up of ice 1 km in radius, with mass loss rate

of 108 gm/sec and a period of 5 years. If the comet is assumed to be

active for about a year at every revolution, how much does the nucleus

shrink per revolution? What is the lifetime of the nucleus?

3. A comet of radius 1 km is moving with an orbital velocity of 60 km/sec

at 1 AU. Calculate the amount of out-gassing required if the gas comes

out at 1 km/sec from the sun-lit hemisphere of icy nucleus so that the

non-gravitational forace is about 1% of the orbital force.

4. How do you reconcile the fact that the nucleus contain mostly water

but the observed geometrical albedo for Comet Halley is 0.04.

5. If the nucleus of a comet was made up of 80% CO2 and rest is made up

of other molecules like H2O, CO and others, what will be the scenario

of such a comet?

6. What are the consequence if the nucleus is a hard solid body instead

of fragile and fluffy in nature?

References

The following papers cover morphology of comets

1. Brownlee, D.E. et al. 2004. Science, 304, 1764.

2. Britt, D.T., Boice, D.C., Buratti, B.J. et al. 2004. Icarus, 167, 45.

3. Thomas, P.C., Viverka, J., Belton, M.J.S. et al. 2007. Icarus, 187, 4).

The following paper gives a good account of the vapourization theory

4. Delsemme, A.H. 1982. In Comets ed. Wilkening, L.L. Univ. Arizona

Press, Tucson, p. 85.

A good account for later work is given in the following papers:

5. Meech, K.J. and Svoren, J. 2005. In Comets, Eds. M.C. Festou, H.U.

Keller and W.A. Weaver, Univ. Arizona Press, Tucson, p. 317.

6. Prialnik, D., Benkhoff, J. and Podolak, M. 2005. In Comets, Eds, M.C.

Festou, H.U. Keller and W.A. Weaver, Univ. Arizona Press, Tucson,

p. 359.

A good summary of albedo and radius of comets is given in the following

paper:



7. Lamy, P.L., Toth, I., Fernandez, Y.R. et al. 2005. In Comets, eds.

M.C. Festou, H.U. Keller and W.A. Weaver, Univ. Arizona Press, Tuc-

son, p. 223.

The mass and density is covered in the following papers.

8. Davidson, B.J.R. and Gutierrez, P.J. 2006. Icarus, 180, 224.

9. Richardson, J.E., Melosh, H.J., Lisse, C.M. et al. 2007. Icarus, 190,

357.

10. Weissman, P.R., Asphang, E. and Lowry, S.C. 2005. In Comets II,

eds. F.C. Festou, H.U. Keller and W.A. Weaver, Univ. Arizona Press,

Tucson, p. 337.

First direct determination of the surface gravity of Comet Tempel 1, is in

the following paper.

11. A’Hearn, M.F., Belton, M.J.S., Delamere, W.A., et al. 2005. Science,

310, 258.

The rotation of the nuclei of comets is covered in the following paper

12. Samarasinha, N.H., Mueller, B.E.A., Belton, M.J.S. et al. 2005. In

Comets, eds, M.C. Festou, H.U. Keller and W.A. Weaver, Univ. Ari-

zona Press, Tucson, p. 281.

The following papers refer to nucleus composition

13. Huebner, W.F. and Benkhoff, J. 1999. Space Sci. Rev., 90, 117.

14. Huebner, W.F. and Benkhoff, J. 1999. Earth, Moon and Planets, 77,

217.

For structure, outburst etc. of cometary nuclei, see papers

15. Belton, M.J.S., Thomas, P, Veverka, J., et al. 2007. Icarus, 187, 332.

16. Prialnik, D., Benkhoff, J. and Podolak, M. 2005. In Comets, eds. M.C.

Festou, H.U. Keller and W.A. Weaver, Univ. Arizona Press, Tucson,

p. 359.

The existence of non-gravitational force in comets was proposed for the first

time in a classic paper by

17. Whipple, F.L. 1950. Ap.J. 111, 375.

For a good summary of non-gravitational forces in Comets, refer to:

18. Yeomans, D.K., Chodas, P.W., Sitarski, G. et al. 2005. In Comets,

eds. M.C. Festou, H.U. Keller and W.A. Weaver, Univ. Arizona Press,

Tucson, p. 137.

The first determination of Ortho-to-Para ratio is in the following papers:

19. Mumma, M.J., Weaver, H.A. Larson, H.P. 1986. Science, 232, 1523;

Astron. Astrophys., 187, 419.

For later work, refer to the following papers:


Nucleus 379

20. Bonev, B.P., Mumma, M.J., Villanueva, L. et al. 2007. Ap.J., 661,

L97.

21. Kawakita, H., Russo, N.D., Furusho, R. et al. 2006. Ap.J., 643, 1337.

22. Kawakita, H., Jehin, E., Manfroid, J. et al. 2007. Icarus, 187, 272.

A good account of binary systems is in the following paper

23. Noll, K.S. 2006. In Asteroids, Comets and Meteors, eds. D. Lazzaro, S.

Ferraz-Mello and J.A. Fernandez, Cambridge University Press, Cam-

bridge, p. 301.


CHAPTER 12

Origin

There are many ideas and hypothesis with regard to the origin of comets.

Some of these date back to Laplace and Lagrange. Dynamical simulations

in conjunction with available observations, although scarce, have given new

insights into this area of study. Some of these aspects will briefly be dis-

cussed in this chapter.

12.1. Evidence for the Oort Cloud

During early times, the emphasis in the study of comets was mainly

on the determination of their orbits. This resulted in the accumulation of

a lot of data with regard to the orbital characteristics of comets, which

gave an insight into some of the physical problems. One such fundamental

aspect, which came out of these studies - namely, nongravitational force has

already been discussed in the previous chapter. Another significant result

refers to the problem of the origin of comets themselves. Several general

characteristic properties can be noted just from the stability of comets.

The long-period comets, mostly in parabolic orbits, appear to come from

all directions in the sky. The short period comets, have low inclination to

the ecliptic plane and have a strong association with the planetary system.

Any reasonable theory of the origin of comets has to explain these features.

Various concepts, ideas, hypothesis and theories, have been put forward

over the years discussing the merits and demerits of interstellar vs solar

system origin of comets. Therefore the whole subject was in a confused

state. In a classic paper, Oort in 1950 proposed the unification of the

origin of comets within a reasonable and consistent framework. This has

given rise to a rapid development of the subject and a better understanding

of the whole phenomena.

381



Table 12.1 Distribution of original values of

semimajor axes, a(AU).

1/a Number of comets

≤ 0.00005 10

0.00005 - 0.00010 40.00015 - 0.00015 1

0.00020 - 0.00020 1

0.00025 - 0.00050 10.00050 - 0.00075 1

> 0.00075 0

Oort, J. 1950, Bull. Astr. Inst. Netherlands

11, 91.

Even with limited observational material available on about 19 long-

period comets at that time, Oort showed that a simple plot of the number

of comets versus the inverse of semi major axis, 1/a (equivalent to orbital

energies) of the original orbit gave a conspicuous peak near value zero (i.e.

nearly parabolic orbits) (Table 12.1). The original orbit refers to the orbit

of the comet before it enters the planetary system.

The peak observed in the 1/a distribution cannot be due to chance but

represents the real characteristic property. This is based on the fact that

the observed dispersion in 1/a is much smaller than it would have been

if it had passed through perihelion passage once. Even one passage can

bring in a dispersion in 1/a which is much larger than the observed values.

Therefore most of the comets must have come into the solar system for

the first time, generally called ‘new’ comets. Most of these comets appear

to come from the region of say 30,000 to 50,000 AU. This led Oort to

recognize the existence of a spherical cloud of comets around the Sun at

this distance but still gravitationally bound to it. This is generally called

the “Oort cloud”. It should be pointed out that as far back as 1932, Opik

had suggested the possibility of the presence of such a cloud surrounding

the solar system. With more accurate and high precision data available

in recent years on a large number of comets the idea proposed by Oort

has been confirmed and all the results deduced earlier based on the limited

data remain more or less the same. It is remarkable that such an important

work came out, based just on 19 cometary orbits. Figure 12.1 shows the

histogram of the number of comets versus the original 1/a values based on

comets with well-determined orbits. The observed distribution has a sharp


Origin 383

peak of comets at near zero. The figure also shows a continuous distribution

of tightly bound orbits. Most new comets have aphelion distance of around

50,000 to 100,000 AU. The corresponding major axis is around 25,000 to

75,000 AU. More recent determination give a value for the semi major axis

of about 36,000 AU. These distances are almost comparable to the distances

of nearby stars. Oort recognized the fact that the peak in Fig. 12.1 should

be the source of long period comets, a gravitationally bound spherical cloud

of comets at distances greater than 104 AU from the sun. Recent simulation

studies have shown that the inner part of the Oort cloud at distances of

around 3000–10,000 AU may also be the source of long period coments.

Fig. 12.1 A histogram of the number of comets plotted as a function of the originalinverse semi major axes for the observed long-period comets. The sharp peak of comets

near zero value of original 1/a represent the new comets from the Oort cloud. (Marsden,B.G., Sekanina, Z. and Everhart, E. 1978. A.J., 83, 64).

Oort made an estimate of the total number of comets in the Oort cloud

to be about 1.9 × 1011 based on the observed flux of dynamically new

comets for an influx rate of about 1 comet per year within 1.5 AU from the

sun. More recent dynamical models gave a value for the number of comets

in the Oort cloud to be in the range of (0.4 to 1.3) × 1012. The present day

mass for comets with a > 20,000 AU and H (absolute magnitude) < 11 is

around 7M⊕ and the total present day mass of the Oort cloud is estimated



to be around 38M⊕.

Cometary fading

It is observed that the number of returning comets after the Oort cloud

spike decreases more steeply compared to the expected rate based on mod-

els which takes into account planetary and stellar perturbations. This is

generally known as ‘Cometary Fading’. Dynamical model calculations have

shown that this effect cannot be attributed to modelling effect, but must

be due to the intrinsic property of the comet. Three possibilities have been

suggested. They are ‘disruption or splitting’ process due to low tensile

strength of the nucleus, loss of volatiles called ‘Extinct State’ or forma-

tion of non-volatile mantle on the surface of the nucleus called ‘Dormant

State’ of the comet. Dynamical calculations once again seems to rule out

the extinct and dormant state of the comet for the observed cometary fad-

ing. Therefore more likely that the catastrophic splitting of comets is the

dominant physical loss process for the observed cometary fading. There-

fore comets after few passes through the solar system are split up into a

large number of smaller bodies which then move independent of each other.

Such a process might have resulted in well known Kreutz group of comets

containing more than few hundred comets.

12.2. Evolution and Properties of Oort Cloud

The transformation of comets from the Oort cloud to the observable

comets can be understood in a quantitative fashion in the following man-

ner. Due to stars perturbation, many comets leave the cloud for ever and

some others enter the planetary system. Among these some may happen to

come close and are observable as “new” comets. When a fraction of these

comets encounter the planets, particularly Jupiter, they are perturbed and

leave the system altogether after their first encounter. Some of these get

caught in the solar system, and are seen as long-period comets. By repeated

encounters with the giant planets, enough of them are progressively trans-

ferred from long-period orbits to intermediate-period orbits and finally into

short-period orbits. A large number of investigations have been carried out

for an understanding of some of these problems through analytic, semi-

analytic and numerical calculations. With the availability of high speed

computers, the Monte Carlo approach has been used to simulate the dy-

namical evolution of comets in the Oort cloud, wherein it is possible to


Origin 385

investigate the effect of varying initial conditions. The idea is to consider a

large number of comets in the Oort cloud and study their time evolution by

taking into account the perturbations due to various sources. The orbits of

comets could be followed until they escape from the system in a hyperbolic

orbit. It is the cumulative effect of the changes in the comets orbital energy

during each passage through the planetary system which is important in

deciding the fate of the final orbit of the comet. Due to large computer time

involved, some approximations have to be made to obtain results within a

reasonable length of time.

The results of Monte Carlo simulation for 105 hypothetical comets gives

a good match to the observed distribution in 1/a. (Fig. 12.1). On the av-

erage a long-period comet with perihelion distance < 4 AU makes around

five passages through the planetary region before taking on some final state

with a mean life time of about 6 × 105 years. In addition to the perturb-

ing forces normally considered, arising due to planetary encounters and

passing stars that continually change the orbital elements of comets, other

type of perturbing forces have been shown to be important. One is due

to the existence of Giant molecular clouds with a typical mass ∼ 3 × 105

M and radius ∼ 20pc respectively. The Giant Molecular Cloud encoun-

ters are rare with the mean interval of about 3 to 4 × 108 yr. This has

been shown to be a major perturber of the orbits of comets in the Oort

cloud. Just as stars passing the Oort cloud perturb the orbit of comets,

in a similar way the distribution of stars in the Galaxy can also have a

major perturbing effect on the orbits arising due to ‘tidal’ distortions. This

arises due to the gradient of the gravitational force in the solar neighbor-

hood which is slightly different at the position of the Sun and the comet.

As a result the galactic acceleration is different at the two locations which

leads to a net tidal force acting on the comet with respect to that of the

Sun. This effect has been taken into account in a more realistic manner

in recent dynamical calculations. The relative importance of various per-

turbers for the net influx of comets can be seen from Table 12.2. The

results are for a radial distribution of comets in the Oort cloud of the form

n(r) ∝ r−3 and normalized to unity for the influx rate caused by random

passing stars. For ranges around 7 × 103 ≤ aoriginal ≤ 4 × 104 AU of

the Oort cloud the effects of forces other than planetary perturbations are

important.

The structure and stability of the Oort cloud has also been studied

based on the N-body simulation with the inclusion of various perturbations.



Table 12.2 Influx rate of comets from

the Oort clouds.

Perturber Relative

number

Random stars 1.0

Intermediate size 124

molecular cloud

Vertical galactic tidal force 1.72

Close stellar passage 62.1(D = 104 AU)

Adapted from Fernandez, J.A. and

IP.W.-H. 1991. In Comets in the Post

Halley Era. eds. C.L. Newburn et al.Kluwer Academic Publishers. p. 487.

This approach tries to simulate the actual physical system, wherein the

interaction of all the bodies is taken into account. It is possible to study

physical processes through this approach, such as the exchange of energy

and angular momentum between the cloud and the stars and so on. A

comparison of the mean square velocity of the cloud with the escape velocity

at any particular layer would show whether it can escape from the system.

The results show that the inner regions are stable, while the outer layers

can escape from the system.

The dynamical half-life of comets in the Oort cloud due to the effect of

passing star is estimated to be about 3 Gyr at 25,000 AU. It is around 1

Gyr at 50,000 AU. The effect of Giant molecular cloud on the Oort cloud is

comparable that of passing star. The resultant effect of stellar perturbation

is that, around 5% of comets should survive at 50,000 AU for 4.5 Gyr.

Similar is the case at 30,000 AU. This indicates that the comets in the

outer region of the Oort cloud decrease with time. One way to replenish

is by capture of comets from interstellar space. This is a highly unlikely

process as it involves a three body gravitational interaction for dissipating

the excess hyperbolic energy. The other possibility being that the supply

of comets is provided by the inner regions of the Oort cloud. However it

is not at all clear at the present time whether the comets are depleted at

such a rate from the outer regions of the Oort cloud that it requires to be

replenished in view of several simplifying assumptions made in the model

calculations.


Origin 387

Fig. 12.2 The results of Monte Carlo simulation showing the number of new long-period

comets entering the terrestrial planets region q < 2 AU from the Oort cloud. The spikesrepresent the comet showers arising due to random stars penetrating the Oort cloud

(Courtesy Heisler, J).

Cometary Showers

The studies based on Monte Carlo calculations have indicated that in-

flux of cometary flux from the outer regions of the Oort cloud could vary

roughly by a factor of 2 to 3. These are a sort of common phenomena.

Sometimes it is possible that the passing stars happen to penetrate the

Oort cloud, or the stars could be of relatively high mass, or pass the cloud

with small impact parameter. In these situations the orbit of comets could

be changed by a larger amount. The net effect will be to scatter more

long-period comets into the solar system, which will increase drastically

the observable comets giving rise to cometary shower (Fig. 12.2). Monte

Carlo simulations have been carried out for such situations to get an idea of

the expected results. The impact rate increases by a factor of around 300

for the case of star passing at a distance of ∼ to 3× 104 AU from the Sun

and seems to last for around 2 to 3 × 106 years. The increase in flux rate

arises due to the fact that comets coming from the inner part of the Oort



cloud have relatively shorter time periods compared to those coming from

outer regions and hence perturbed more due to planetary perturbations

and in turn make more revolutions. However such close stellar encounters

are quite rare and the frequency of occurrence could be perhaps once in 3

to 5× 108 years or so. In a similar way, a sudden increase in the influx of

comets from the Oort cloud can increase by a factor as high as 103 for a

close encounter with a Giant molecular cloud (M ∼ 5 × 105M) or even

higher rate of influx for the penetrating Giant molecular cloud. But such

occurreness may occur at intervals of several times 107 years.

The distribution of comets is a complicated function of the various phys-

ical mechanisms which could operate on the Oort cloud. Therefore the

enhancement of ejection of comets could vary in a periodic fashion, ran-

dom or in some complicated way, depending on the manner in which the

disturbances act on the Oort cloud. This could have catastrophic or mod-

erate effects on the geophysical, biological or climatic changes on the earth.

They could possibly appear in terrestrial records. In view of these consid-

erations, it has been suggested that the random cometary showers could

contribute substantially for the formation of craters on planets and the

Moon and also the biological extinction events on the Earth. Also the sta-

tistical study of the fossil records has been interpreted in terms of biological

extinction events that may be taking place with a time scale of approxi-

mately 26× 106 years (Fig. 12.3). An interesting measurement was carried

out at the sea floor looking for iridium concentration for the period from

33 to 67 Myr ago. The results showed an increase by a factor of around 13

near the Cretaceous-Tertiary boundary (referring to 66 Myr). This can be

explained in terms of a strong comet shower which enhanced the iridium

concentration. The expected value from such a process agrees within a fac-

tor of two of the observed value seen at the Cretaceous-Tertiary boundary.

At present these may be considered as possibilities rather than evidence in

support of them, as more work needs to be done from both observational

and dynamical points of view.

The Oort cloud has been considered to be a cold storage place for long-

period comets without much happening until it is perturbed by a perturbing

force. In recent years, it has been suggested that during the life time of

the Oort cloud the outer layers of the cometary nuclei could be modified

due to various physical processes such as, bombardment by galactic cosmic

rays, solar wind, heating due to nearby supernova explosions, luminous

stars passing by, impact due to various debris and so on.


Origin 389

Fig. 12.3 A statistically significant mean periodicity of ground 26 million years is indi-cated by the extinction records for the past 250 million years. (Raup, D.M. and Sepkoshi,

J.J. 1984. Proc. Nat. Acad. Sci. 81, 801).

12.2.1. Short period comets

The long-period comets, after random-walk through the planetary sys-

tem, which reduces the semi-major axis of the orbit, could finally give rise

to short period comets. The important work of Everhart in 1972 showed

clearly that such a dynamical mechanism can be an efficient process. The

question to ask is whether such a mechanism can produce the number of

observed short-period comets.

Estimates of the expected number of short period comets from long

period comets by diffusion process has been made and it varies consider-

ably. The dynamical calculations have shown that the ratio of the initial

population of near parabolic orbits to the number of long period comets

gravitationally bound to the system after N perihelion passages varies as

N1/2. The ratio of the observed long period comets to short period comets

is ∼ 5 and hence N ∼ 25. This indicates that after around 25 passages or

so, around 90% of the original comets are lost from the system. It is quite

possible that some of the long period comets can be transformed into short

period comets after an unusually long number of revolutions. It is estimated

that around 2× 103 revolutions are required for transforming a long period

comet to short period comet with P < 20 years. The transformation can



also take place only over a restricted region, 4 < q < 6 AU and 0 < i < 9,

with an efficiency factor of one captured comet for about 102 long period

comets. In addition, other physical processes such as sublimation, breaking

of the nucleus and so on could also prevent the long-period comet turn-

ing into a short-period one. Therefore the existing dynamical calculations

show that roughly one comet out of a total of 103 comets can end up as a

short-period comet. Hence the major difficulty in the process of producing

long-period comets to short-period comets through diffusion process in the

solar system is that the resulting number of short-period comets are too

small compared to the observed number. Also the long-period comets tend

to preserve their inclination as they evolve into short-period comets, in con-

trast to those of observed short-period comets, whose orbits are confined

to inclinations <∼ 30 or so. In addition, the original population of long

period comets with random inclinations would lead to a certain fraction of

short period comets with retrograde orbits, which is not seen. These are

some of the major difficulties and therefore it is necessary to consider other

mechanisms for the production of short period comets.

In order to produce low inclination, short period comets, the source

has also to be confined to a flattened disk of comets. This has led to the

hypothesis that the Trans-Neptunian population of comets could be the

source of short period comets. The possibility of the existence of such a

source of comets had been suggested by Kuiper in 1951. He postulated a

remnant of the accretion disk of planetesimals in the solar nebula, which

never managed to accrete into planets. This is now termed as ‘Kuiper Belt’.

The Kuiper Belt is also called ‘Edgeworth-Kuiper Belt’ as Edgeworth had

shown the possible presence of comets beyond pluto in 1949. This idea

has been elaborated later on by several investigators and showed that such

a residual population of planetesimals could have remained bound to the

solar system. The short period comets produced from such a comet belt

are far more efficient than produced through long period comets from the

Oort cloud.

Detailed simulation studies have shown that Oort cloud is composed of

two parts, namely the spherical outer cloud and the inner cloud which is

flattened. The boundary between the two regions is around 20,000 AU.

Long period comets come from the the spherical distribution of the Oort

cloud, Halley type comets come from the inner flattened disk of the Oort

cloud and Jupiter family of comets come from the Kuiper belt. It now ap-

pears likely that most Jupiter family comets arise from the related structure

called the scattered disk.


Origin 391

Extensive dynamical evolution studies over forward and backward in

time spanning up to 107 yr has been carried out. The comet was followed

until it became either an unbound system from the sun or a sun-grazer.

These studies show that around 92% of comets were ejected from the solar

system and around 6% of comets were destroyed by becoming sun-grazers.

It also showed that around half of the comets of sun-grazing type were of

Halley type of comets. The median life time of short period comets ∼4.5

× 105 yr and that of JFCs ∼3.2 × 105 yr. The distribution of comets

based on simulation studies is in agreement with the observed distribution

of Jupiter family of comets. These studies are consistent with the earlier

work in showing that Kuiper belt is the source of JFCs. Simulations also

indicate that JFCs and HTCs switch between them in about a dozen times

in a time scale of 4.5 × 105 yr.

Since the Kuiper Belt is believed to be the source of short period comets

(i.e. Kuiper Belt objects, KBOs) considerable effort went into looking for

KBOs and the first such object was discovered in 1992. Since then, large

number of transneptunian objects (KBOs) have been detected. The esti-

mated number of KBOs of sizes ∼100 km with 30 < a < 50 AU is about

105. Simulation studies for the dynamics of KBOs show that the orbital

characteristics of the scattered KBOs that enter into the solar system are

quite similar to those of Jupiter family comets.

12.3. Origin of the Oort Cloud

The reservoir of comets generally called the Oort cloud is quite well

established. The observable long-period comets can be understood reason-

ably well, based on the time evolution of new comets coming out of the

Oort cloud. Therefore the origin of long-period comets essentially reduces

to the problem of the origin of the Oort cloud. Various hypothesis have

been proposed to explain the origin of the Oort cloud. They can all be di-

vided roughly into two groups, interstellar and the solar system. The other

possibility of comets forming at the present location of the Oort cloud itself

is not feasible due to the extremely low densities, which make it difficult

for the growth of the particles.

If interstellar comets do exist and enter the solar system at times then

they should be seen with hyperbolic orbits. However, no such comet has

been observed for the last 200 years or so. This observation can be used to

put a limit on the number of interstellar comets in the solar neighborhood



to be ≤ 10−4 AU−3. This shows that the number of interstellar comets

entering the solar system at the present time is not large. In addition,

capturing hyperbolic comets from interstellar medium is a highly unlikely

process as a third body is required to dissipate the excess kinetic energy

involved. It is quite possible that passage of the solar system through

a Giant molecular cloud or dense region might result in the capture of

comets leading to bound orbits around the Sun. One can also think of

other scenarios which could have given rise to the Oort cloud. Although

these are possible in principle, however there appears to be no concrete

viable models based on such mechanisms.

The other possibility that comets is of solar system in origin implying

that they were formed at the same time and as a part of the formation of

Sun and planets has received considerable interest and attention. On this

hypothesis, cometary material should be similar to that of the solar system

material. This is borne out by the measurement of the isotopic ratio of12C/13C in comets ∼ 100, comparable to the terrestrial value of 89.

However, the formation of comets in the inner solar system is highly

unlikely due to the fact that the presently known chemical composition of

comets requires a temperature at the time of formation to be quite low<∼ 100 K to keep the volatiles like H2O, CO2, CO, NH3 and CH4 from

evaporating. This led to the other possibility that the comets were formed

in the outer parts of the nebula that formed the planets. The chemical

composition of the solar system bodies can roughly be divided into three

classes depending upon the characteristics of the elements present in them.

For example, gases such as hydrogen, helium and noble gases stay as gas

even at low temperatures, ice melts at moderate temperature and lastly

the terrestrial materials like silicon, magnesium, and iron melt at higher

temperatures. Table 12.3 shows the relative abundances of various ele-

ments in the solar system bodies. It is clear that Jupiter and Saturn were

formed mostly of the original solar material like the Sun, while Uranus,

Neptune and Comets were formed in the colder regions which account for

the icy material. Therefore, probably comets were formed beyond Sat-

urn.

Additional evidence for the low formation temperature of T ≤ 35 K,

comes from the observation of S2 in comets. Additional support comes

from the study of ortho and para ratio of H2O, NH3 and CH4 in comets

which indicate the temperature of formation of ice ∼ 35 K. Thus there are

several observational results which seem to show that comets are formed


Origin 393

Table 12.3 Relative abundances of atoms by mass in the solar system.

Elements Sun Terrestrial Jupiter Saturn Uranus, Neptune

planets and and comets

meteorites

Gaseous H 1.0 Trace 0.9 0.7 Trace

He

CIcy N 0.015 Trace 0.1 0.3 0.85

O

MgEarthy Si 0.0025 1.0 Trace Trace 0.15

Fe

etc.

Whipple, F.L. 1972. In Motion, Evolution of Oribits and Origin of Comets eds.Chebotarev, G.A. Kazimirchaka-Polonskaya, E.I. and Marsden, B.G. Dordrecht:

D. Reidel Publishing Company, p. 401.

far from the Sun and in the cooler regions of the solar nebula, placing them

beyond Uranus-Neptune zone, possibly the Kuiper belt.

In this connection it is interesting to note that Oort had proposed early

on that asteroid belt could be the source region for the long-period comets.

It was Kuiper who was the first to propose that due to the icy nature of

the comet, they are more likely to be formed in the region of orbits of giant

planets, a concept that has withstood over time.

The ejection of minor bodies appears to be a natural consequence of the

accumulation process of giant planets. Due to gravitational perturbations

of various planets, the orbit of some of these bodies can detach themselves

from the bound system. Monte Carlo simulations have demonstrated clearly

that such mechanisms can be present in the system. The results of such

calculations have shown that the planets tend to eject icy planetesimals

from their present zones to eccentric long period orbits with semi-major axis

≤ 103 AU. At these distances the perturbations due to stellar and galactic

sources tend to be important to push the orbits further into the region of

the Oort cloud and thus detaching them away from the zone of influence

of planetary perturbations. This can finally lead to a cloud with more

comets at its centre compared to its outer region, similar to the Oort cloud.

A typical result of the Monte Carlo calculations for the assumed initial

hypothetical comet distribution confined to the ecliptic plane is shown in

Fig. 12.4. They show the expected distribution in the plane perpendicular

to the plane of the Galaxy for different times. The comets are more or less



randomized both in eccentricity and inclination in a time period of around

4.5 × 109 years and for distances beyond around 104 AU. It is also found

that relatively more icy planetesimals are ejected out of Uranus-Neptune

region compared to Jupiter-Saturn region. The dynamical calculations,

thus indicate that the Oort cloud can be produced as a natural by-product

of the various processes that could take place in the solar system during its

lifetime.

Fig. 12.4 The results of dynamical evolution of a cloud of comets ejected from Uranus-Neptune zone are shown projected onto a plane perpendicular to the galactic plane for

several time intervals. The dotted circle with a radius of 2× 104 AU separates the innerand outer Oort clouds (Duncan, M., Quinn, T. and Tremaine, S. 1987. AJ, 94, 1330).

The best way to study the structure of the Oort cloud is to directly

image the comets and make a statistical analysis. This is still a long dream

as the object would have a magnitude ∼42 in the inner region ∼ 3000 AU

and ∼ 60 magnitudes at around 20,000 AU in the Oort cloud that it will not


Origin 395

be possible to detect them in the forceable future. It is possible in principle

to detect objects in the inner regions of the Oort cloud through occultation

of stars. This is again a long term goal. Therefore the only method that is

available at the present time is to study the orbital distribution of known

comets for inferring the structure of the Oort cloud. This is again a difficult

proposition as many new comets may not come within 10 AU of the sun

due to Jupiter, which can deflect many of them.

Until a few years ago the detection of objects in the Kuiper belt (∼40 AU) was a distant dream. But not any more as several hundred objects

known as Kuiper belt objects have been detected. In the same spirit the

objects will be looked for in the inner region of the Oort cloud with the

hope of finding some in the not too distant future.

Although the presence of the Oort cloud is well-founded, questions per-

taining to its origin, physics, dynamics, structure and stability are still

uncertain. One method to get some idea of the age of the cloud is through

the study of the isotopic ratios of elements. The 12C/13C ratio available

for comets indicates a ratio of about 100. This ratio is more like the solar

system value (89) rather than the average value of about 40 to 80 of the in-

terstellar material (Chap. 5). This seems to show that comets were formed

out of the solar nebula material about 5 billion years ago.

Dynamical studies which take into account the various perturbations in-

dicate that the Oort cloud can account for the observed long period comets.

The perturbation of the Oort cloud due to the encounter with Giant molec-

ular cloud and galactic tidal force is found to be important, in addition to

planetary encounters and passing stars. The passage of a perturber close

to the Oort cloud can give rise to shower of comets. It can have severe

effect on the Earth’s environment, as revealed by the interpretation of geo-

physical, biological and terrestrial records. The formation of short period

comets from long period comets through diffusion process in the solar sys-

tem has some difficulties, including accounting for the observed number

of short period comets. This has led to the hypothesis that the source of

short period comets may lie beyond the orbit of Neptune between 30 and

50 AU, probably the remnant of the non-accumulated material. This region

is generally known as Kuiper belt. As ejection of minor bodies appears to

be a natural consequence of the accumulation process of Giant planets, this

process could have been responsible for accumulation of comets during the

lifetime of solar system in the region of space, where one now finds the Oort

cloud.



12.4. Taxonomy

The general scenario that emerges from the study of origin of comets

is that, icy planetesimals (comet nuclei) reside mostly in the two regions

of the solar system, namely Kuiper - Edgeworth Belt and Oort cloud(OC).

The OC comets are the ones ejected from the giant - planet region of the

solar nebula (∼5–30 AU). OC comets include long period comets and dy-

namically new comets. The Jupiter-family of comets (short period and

Halley-type, JFCs) originate from Kuiper-Edgeworth Belt (>30 AU). Dy-

namical studies indicate that comets present in the Oort cloud originated

roughly in equal proportion from each giant planet and the Kuiper belt.

These studies have also shown the possibility of radial transport of material

inward and outward of giant - planet region during the formation period of

comets. During this time the nebular temperature was around 200 K near

Jupiter (∼5 AU) and around 30 K near Neptune (∼30 AU). These studies

indicate that JFCs should have been formed at lower temperature environ-

ment compared to those of long period comets. Therefore there would be

variation in chemical composition among these classes of comets as well as

differences in the same group. Therefore the study of the chemical composi-

tion of comets should give information on the proto-solar nebula chemistry

and in turn on the processes that could have taken place in the early solar

nebula.

The study of the molecular composition of volatiles in comets has helped

a great deal in this regard. The molecular composition of a large number

of comets derived from observations in the radio region has clearly shown a

wide variation in the observed chemical composition among comets. Table

12.4 give the average volatile composition of five OC comets which dif-

fer substantially from that of OC Comet 1999S4(LINEAR). The Comet

C/1999S4(LINEAR) is really an unusual comet. It is possible that the five

comets which have similar composition, except CO and CH4, were formed

in the low temperature region, probably formed beyond 30AU from the sun,

while Comet S4(LINEAR) could have formed around 5–10 AU.

Extensive compositional surveys have been carried out on a large num-

ber of comets. One such survey relates to the study of the production rates

of C2, C3, OH, NH and CN in a homogeneous data set of 85 comets. It con-

tained 39 Jupiter-family comets 8 Halley-type comets 8 dynamically new

comets and 27 long-period comets. Although the overall chemical com-


Origin 397

Table 12.4 Composition of Oort cloud comets.

Comets CO CH4 C2H2 C2H6 HCN CH3OH

Average of 1.8-17 0.5-1.5 0.2-0.3 0.6 0.2-0.3 25 cometsa

LINEAR 0.9 ±0.3 0.18 ±0.06 <0.12 0.11±0.02 0.10±0.03 <0.15

a: Comets Lee, Hale-Bopp, Hyakutake, Ikeya-Zhang, McNaught-Hartley

Adapted from Mumma, M.J. et al. 2003, Adv. Space Res., 31, 2563.

position was similar in these comets, there was significant compositional

grouping of comets likely to be related to their place of formation. In partic-

ular, there was a depletion of production rate ratio of C2/CN (carbon-chain

molecules) in a large number of short period comets belonging to Jupiter-

family. This led to the classification of comets into two groups called,

comets with typical mean values of C2/CN ratio (0.06 ± 0.10) and comets

with depleted abundance ratio of C2/CN (-0.61 ± 0.35). The results of

these studies also indicated that approximately one-half of Jupiter-family

of comets belongs to a typical class, while the rest of the comets belong

to the depleted class with varying degree of C2/CN ratio. This has given

rise to a new terminology ‘Comet Taxonomy’ in which comets are distin-

guished based on their carbon-chain production ratio, C2/CN, as typical

or depleted. Most likely the depleted comets originate in the Kuiper belt.

This has direct relevance to the long standing question whether all or part

of Jupiter-family of comets come from the Kuiper belt. The depletion of

C2/CN in some comets is attributed to primordial rather than due to evo-

lutionary effect as no correlation with dynamical age was seen.

There are other types of observations which have direct bearing on the

primordial chemical composition of the solar nebula and in turn on the

formation sites of comets.

Deep Impact studies of Comet Tempel 1 and the studies of Stardust

samples of Comet Wild 2 have provided detailed mineralogical content of

dust particles in these comets. The presence of crystalline silicate and cal-

cium - and aluminium - rich inclusions (CAIs) in the dust particles indicate

that they must have been formed at higher temperatures ∼1400 K or so.

Therefore these dust particles are believed to have been formed in the hot-

ter and hence in the inner regions of the solar nebula. Since these dust

particles came from the icy comet that was formed in the Kuiper belt (out-

skirts of the solar system), this require mixing of material from the inner

regions of the solar system with that of Kuiper belt region.



The enhancement of D,12C13C and 15N13C seen in the dust particles

from comets indicate that cometary material contain presolar material.

Therefore there is also mix of solar system material with that of inter-

stellar and proto-stellar material. All these studies could provide clues to

the place of origin of comets in the solar system. They could also help in

the understanding of the physical processes that led to larger scale mix-

ing in the solar nebula. These studies could in turn provide important

information with regard to the origin and evolution of the solar system.

12.5. Summary

Oort showed from the study of 19 comets, the existence of gravita-

tionally bound spherical cloud of comets at distances larger than 104 AU,

called the Oort cloud. The comets in the Oort cloud are perturbed by

various forces which make them leave the system. Some of them are cap-

tured by the solar system and it can account for the observed long period

comets. The formation of Oort cloud is by the accumulation of icy planetes-

imals ejected from the giant-planet region (∼ 5–30 AU). This is a natural

outcome of the various processes that could have taken place in the solar

system during its life time.

The diffusion process of long period comets in the solar system cannot

explain the observed number of short period comets. Therefore short period

comets is believed to come from a different source (>30 AU) called Kuiper

Belt. Large number of objects in this region, called Kuiper Belt objects

(KBOs) have been seen whose characteristics are similar to those of Juipter

family of comets. Therefore the general picture at the present time is that,

long period comets comes from the spherical distribution of the Oort cloud,

Halley type comets comes from the inner flattened region of the Oort cloud

and Jupiter family of comets comes from the Kuiper Belt. Therefore comets

are formed in the colder regions of the solar system which account for the

icy material. There is also evidence for the large scale mixing of the material

between inner and outer regions of the solar nebula.

Problems

1. How can one say from the appearance of a comet in the sky whether it

is a new comet coming from the first time or whether it is one which

has already been seen?


Origin 399

2. What is Jupiter’s family of comets? What is the most probable origin

of these comets?

3. Is there any evidence that the Oort cloud was formed about 5 × 109

years ago? Can you think of some other process by which the Oort

cloud be of more recent origin, formed some 106 to 107 years ago?

4. Is there any way to test observationally the existence of the Oort cloud?

References

The basic idea of the cloud of comets is from the classic paper

1. Oort, J. 1950. Bull. Astr. Inst. Netherlands 11, 91.

Other relevant paper

2. Kaib, N.A. and Quinn, T. 2009. Science, 325, 1234.

Earlier references to the cometary clouds can be found in

3. Opik, E. 1932. Proc. Am. Acad. Arts and Sciences, 67, 169.

The best book on the origin of Comets is the following:

4. Bailey, M.E., Clube, S.V.M. and Napier, W.M. 1990. The origin of

Comets. Pergaman Press.

A good account of several aspects are given in the following papers:

5. Dones, L., Weissman, P.R., Levison, H.F. et al. 2005. In Comets,

Eds. M.C. Festou, H.U. Keller and H.A. Weaver, Univ. Arizona Press,

Tucson, p. 153.

6. Morbidelli, A. and Brown, M.E. 2005. In Comets, eds. M.C. Festou,

H.U. Keller and H.A. Weaver, Univ. Arizona Press, Tucson, p. 175.

Papers pertaining to KBOs: First Detection

7. Jewitt and Luu, J. 1993. Nature, 362, 730.

The following paper may be referred for a good summary

8. Barucci, M.A., Doressoundiram and Cruikshank, D.P. 2005. In Comets,

Eds. M.C. Festou, H.U. Keller and H.A. Weaver, Univ. Arizona Press,

Tucson, p. 647.

For a discussion pertaining to Taxonomy see also Section 6.6


CHAPTER 13

Relation to Other Solar SystemStudies

Comets appear to have a close relationship with other objects of the

solar system like asteroids and meteorites. The meteor streams which often

appear in the sky are also believed to come out of the cometary material.

The study of the nature and composition of dust particles collected at high

altitudes through rockets and satellites indicates that they could be of ex-

traterrestrial origin and are generally associated with comets. In recent

years comets have also attracted great attention in view of their possible

impact on the life of our planet. This has come about basically from the ob-

servation that the complex molecules of various kinds appear to be present

in the nuclei of comets. This in turn is related to the chemical composition

of the primordial material of the solar system. We would like to elucidate

some of these aspects here.

13.1. Asteroids

The minor planets of the solar system were classified as objects having

dimensions less than Pluto. The Ceres which was discovered in 1801 was

classified as a minor planet. This was followed by the discoveries of other

minor planets called Pallas in 1802, Juno in 1804 and Vesta in 1807 and so

on. However, these objects turned out to be far fainter than expected as

their orbits are closer than Jupiter’s distance. It was realized that these are

of different kind of objects. Hence, minor planets are also called asteroids.

The largest concentration of asteroids is in the range of 2 to 3.5 AU, placing

them between the orbits of Mars and Jupiter. This is generally called the

asteroid belt. There are two other minor concentrations of asteroids known

as Hildas and Trojans which are located at distances of around 4 and 5 AU

respectively. It is estimated that more than around 100,000 asteroids have

401



been seen at least once. The computed orbits are accurately known for

a large number of asteroids. The period for most of the asteroids is in

the range of around 3 to 6 years. They revolve round the Sun in the same

direction as the principal planets. Most of them have orbits which lie nearly

in the plane of the Earth’s orbit and the average inclination is about 10.

The average eccentricity of the orbits is around 0.15 with sharp cut offs on

either side.

Observing the light curve, it is possible to determine the rotation period

and shape of asteroids. The rotation period of most of the asteroids is of

the order of a few hours. For example the rotation period of Eros ∼ 5h

17m and that of Juno ∼ 7h 13m. The deduced mean geometrical albedo

of asteroids is rather low, ∼ 0.05 which implies that they are extremely

dark objects. The diameter of asteroids can vary from the lower set limit of

about 100 meters to about 1000 km as seen in the case of Ceres. Most of the

asteroids are found to be extremely irregular in shape. This indicates that

they were most probably produced out of the break up of some parent body.

In addition, the collisions of asteroids within the belt is quite frequent and

this leads to erosion and fragmentation.

Since 1965, space missions such as, the Mariner Series, the Vikings and

the Hubble space telescope have been used for the study of asteroids. The

two Martian satellites, Phobos and Deimos, two likely captured asteroids

have been studied in great detail. They are found to be approximately

elliptical in shape with radii of 13.5 × 10.8 × 9.4 km and 7.5 × 6.2 × 5.4

km respectively. The mass of Phobos derived from the perturbations of the

spacecraft is around 1.3× 1019 gm. The estimated average density of Pho-

bos and Deimos is around 2.0 gm/cm3 and the geometric albedo ∼ 0.06.

High resolution observations have shown many surface features, including

large craters. More recent spacecraft missions which passed by closer to

asteroids have given detailed information about their shapes, sizes and sur-

face features. Table 13.1 gives a list of some of the spacecraft missions to

asteroids. The NEAR-shoemaker spacecraft after circling the asteroid 433

Eros for a year landed finally on this asteroid. The ultraviolet observations

carried out with satellites on a large number of asteroids show that the ge-

ometrical albedo decreases towards shorter wavelengths. For example, the

geometric albedo for Ceres at wavelengths 4300 and 2600 A is about 0.031

and 0.023 respectively. It is seen that the albedo of asteroids seem to be

almost in the same range as that of cometary nuclei.


Relation to Other Solar System Studies 403

Table 13.1 Spacecraft to asteroids.

Spacecraft Name of the Diamension of Remarks

asteroid the asteroids

(diameter, km)

Galileo Gaspara 19 × 12 × 11 Craters present,

S-type243 Ida Dactyl 56 × 24 × 21 Craters present,

S-type

NEAR-Shoemaker 253 Mathilda 52 Large craters

present, density∼1.3g/cm3

C-type433 Eros 34 × 13 × 13 Craters present

(NEA asteroid) density ∼2.7g/cm3

S-type, landed onFeb. 12, 2001

Deep space 1 9969 Braille 2.1 × 1 ×1 Detected magnatic

field

Stardust 5535 Annefrank 6.6 × 5.0 × 3.4 Appear angular,

flat surfacesHayabusha NEA 25143 0.5 Exhibit rocks

Itokawa and boulders, smoothareas, no impact

craters

Infrared observations have given information about the surface compo-

sition of asteroids through the detection of characteristic bands of minerals.

The relative abundances of silicates like olivine, pyroxene, feldspar and so

on has been determined. Most of them fall broadly into two types with

respect to their chemical composition: the C type, the darkest asteroids

due to the presence of carbonaceous material and the S type which has sil-

icates like olivine and pyroxene. The C type asteroids are more numerous

comprising around 60% of the observed asteroids.

The emission spectra of Trojan asteroid (624) Hektor is compared with

the spectra of cometary dust in Fig. 13.1. This emission spectra is similar

to the spectra of Comets Hale-Bopp and Schwassman-Wachmann.

The origin of asteroids is still not known but they could be of cometary

origin. The short-period comets after many revolutions could become in-

active as most of the volatiles would have vapourized. Depending on the

presence or absence of volatiles, comets are called ‘Dormant’ (the volatiles

are not coming out due to mantle formation) or ‘Extinct’ (when volatiles



Fig. 13.1 Comparison of the emission spectrum in the wavelength range of 5 to 40 µm

of Comets Hale-Bopp and Schwassmann-Wachmann 1 with a Trojan asteroid Hektor.There is similarity between them (Fernandez, Y.R. et al. 2007. In Asteroids, Comets

and Meteors, IAU Symposium 229, Cambridge University Press, Cambridge, p. 121).

have evaporated). Such residual nuclei of comets are indistinguishable from

those of asteroids. There are two other suggestions with regard to the ori-

gin of asteroids and are found in the belt itself. The irregular shape of the

asteroids indicates that they were produced due to the break up of some

parent body at its present location. There is the other possibility that the

material that exists in the present location, is the material that could not

have accreted into a planet due to some reason.

There are a number of asteroids with large orbital eccentricities and

they are found to come regularly to the Earth. Their perihelion lies interior

to Earth’s aphelion and are called ‘Apollo objects’, named after the first

object of this kind to be discovered. There is another class of asteroids

which includes those which have somewhat larger orbits with perihelion of

around 1.02 to 1.3 AU and are called ‘Amor objects’. In general these two

populations are not fundamentally different from each other and are often

called with a single population as Earth-approaching Apollo-Amor objects.

The mean dynamical life time of Earth-crossing objects is ∼ 107 to 108 yrs,

which is short compared to the age of the solar system. Therefore there

should be a constant source of these objects. Dynamical simulations have



shown that the orbit of comets can evolve into asteroid orbits. Therefore

some of the asteroids can be extinct or dormant comets. It is likely that

the Earth-crossing asteroids are old and non-active comets. In fact, the

traditional distinction between asteroids and comets is becoming nebulous

at the present time, as some of the asteroids with comet like orbits could

be dormant comets. There are some asteroids which appear to be parent

objects to some meteor streams, for example the orbit of (3200) Phaeton

coincides with that of the Geminid Stream.

There could be comets which orbit in the main asteroid belt. An in-

teresting example of this type is the Shoemaker-Levy 9 prior to its capture

by Jupiter. Before it was captured by Jupiter, it was located in the Hilda

asteroid zone. There could be other comets, in the Hilda group of asteroids.

The object 2060 Chiron discovered in 1977 was classified as an asteroid.

It has an orbital period of 51 years and eccentricity of 0.38. It has a semi-

major axis of 13.7 AU which lies in between Saturn and Neptune. Suddenly

in 1988, brightness variations were seen and coma was formed and it was

detected. The emission bands of CN were also detected in 1990. Extensive

observations of this object indicated that Chiron is a comet. Calculations

based on the cometary nucleus model as discussed earlier (see Sec. 11.2)

can explain the observed out-gassing and indicated that it will last for a

long period of time.

The asteroid 4015 is an interesting object in which a comet nucleus has

become an asteroidal object. This object was discovered in 1979 and was

classified as an asteroid. When the accurate orbit of this object became

available, it was possible to identify an object during its earlier time in its

orbit. The object which had the same orbit as that of asteroid 4015 and

was identified in the Palomer sky survey images taken in 1949 was actually

a comet and called Wilson-Harrington. It had plasma tail at that time.

This is an example of a comet turning into an asteroid. This is now called

4015 Wilson-Harrington.

The other example is the object 7968 Elst-Pizarro (133/E-P) which

resides in the main asteroid belt but show recurrent comet-like activity,

such as showing temporary dust tail etc. The cause of such activity could

be that volatile regions may be located near one of the rotational poles of the

nucleus. When this region tilts towards sun, it is exposed to solar radiation.

This sublimes volatiles leading to comet-like activities. The cases of such

unusual behaviour could arise from population of asteroid-comet transition

objects. Therefore there may be large number of comets posing as asteroids

in the asteroidal belt. They could be there in dormant or extinct state. It



is difficult to identify these objects. Even though cometary nuclei contain

much more ice than asteroids, it is likely that evolved comets may have

many characteristics similar to asteroids.

These studies seem to indicate that whether the volatiles are present or

not depend upon the dynamical life time, life time for the volatile compo-

nent to disappear and on the processes that could have taken place inside

the nucleus. Hence, depending upon these factors it is possible to have

different situations.

Binaries

The search for satellites of asteroids (if any) dates back to 1801 and it

had been an unsuccessful one. The serendipitous discovery of a satellite

around the asteroid 243 Ida came from the images taken by the Galileo

spacecraft when the spacecraft visited 243 Ida in 1993. The satellite called

Dactyl is about 1.5 km across and orbits Ida at a distance of approximately

85 km with a period of 1.5 days. Both Ida and Dactyl are S - type asteroids.

This led to the detection of satellites around a large number of asteroids.

The advantage of a binary system is that it is possible to derive the system

mass from the observation of their orbits. But in practice it is rather

difficult to calculate the orbits because the data is limited. Hence the

orbits and the masses have been determined for a small number of binary

asteroids. For example, the estimated mass of (87) Sylvia, the main belt

asteroid, is around 14.78 × 109 kg and that of 243 Ida is around 0.042

× 109 kg. In the main belt the dominant process for the formation of

binaries is through collisions. Numerical simulations have shown that after

collision the fragment may be captured or the fragment may be re-accreted

around the remnant primary. The asteroid Sylvia has been found to be

a triplet system. Numerical simulations have shown that collisions can

produce triplets as well as binaries.

13.2. Meteorites

The meteorites are extra-terrestrial bodies which survive the passage

through Earth’s atmosphere and fall to the ground. This is generally ac-

companied by a flash of light in the Earth’s atmosphere like that of a meteor.

Only the largest objects of some several tens of meters in diameter create

large impact craters when they hit the ground. Large number of meteorites

are known at the present time. Relatively more meteorites are found in



Antartica as they are preserved over long periods of time. It is possible

that due to the fragmentation of the body as it passes through the Earth’s

atmosphere, it may be scattered over hundreds of kilometers on the Earth.

Since early times, meteorites have been studied extensively as they were

shown to be extra-terrestrial in origin. Also, these are only extra-terrestrial

objects that were available for laboratory investigation for a long time.

Large number of meteorites have been studied over the years with regards

to their structure, morphology, chemical composition, origin etc. The two

meteorites of particular interest are the Murchison and Allende meteorites.

Murchison meteorite fell near Murchison in Australia in September 1969.

Allende meteorite fell near Pueblito de Allende in Maxico in February 1969.

Since Murchison and Allende meteorites were very large in , large amount

of material were available for laboratory investigation. Hence extensive in-

vestigations of various kinds were carried out in the laboratory on these two

meteorites. The great advantage being the material from these meteorites

were picked up soon after their fall and hence the contamination is minimal.

A detailed analysis of the chemical and mineralogical composition of

several meteorites indicate that they can broadly be classified into three

groups. They are iron, stony-rion and stony meteorites (Table 13.2). The

iron meteorites are essentially pieces of metal containing Fe-Ni alloy. The

Stony-iron contains mixture of stony material and metallic iron. The stony

meteorite consists of mostly silicate type of material (They are generally

mistaken for terrestrial rocks). Stony meteorites are further classified into

two types, chondrites and achondrites. The most common form of stony

meteorite is the ordinary chondrite in which mm size spheroidal globules

called chondrules are present. Chondrules are essentially silicate mineral

assemblages which had an independent existence prior to the formation of

the meteorite. The sub-group in chondrites is called carbonaceous chon-

drites which are darker stones containing relatively high carbon content.

They are also well known for having mm sized refractory inclusions (Ca-

Al-rich inclusions). The mineralogy in dominated by oxides and silicates.

The stony meteorite without chondrites are called achondrites and are rel-

atively rare. The chondrites have a composition which is similar to that of

the photosphere of the sun, for all but the most volatile elements. This sug-

gest that chondrites are the materials that formed at the same time as the

sun from the solar nebula and since then, they have undergone very min-

imal chemical modification. The idea of primitive character of chondritic

meteorites is also strengthened by their radiometric age, which is roughly

the age of the solar system. On the other hand, achondrites differ in com-



position considerably from that of the sun. The percentage fall of these

types of meteorites are around 94% stony (chondrites, 86%; achondrites,

8%), 5% iron and 1% stony-iron.

Table 13.2 Main meteoritic groups.

Group Main Characterization Main Components

Iron meteorites More than 90% metal Nickel-Iron

Stony meteorites: More than 75% stony material

(i) chondrites chondrules Silicates, solarin composition except

volatile elements

(ii) Achondrites No chondrules Silicates, differ

from solarcomposition

Stony-Iron meteorites:

(i) Pallasites Olivine in Fe-Ni Olivine, metalalloy network

(ii) Mesosiderites silicate and metal silicate, metal

grown together

Meteorites are classified into two groups depending on their presumed

origin. They are called as Primitive meteorites and Differentiated mete-

orites. The primitive meteorites are not subjected to high temperature and

high pressure environment. Therefore the material remain the same from

the time of formation. However differentiated meteorites contain highly

processed material as a result of internal heating. They could also be the

fragments of differentiated parent bodies. Most of the stony meteorites are

primitive. Iron and stony-iron are differentiated meteorites. Most likely,

the iron must have originated from the metallic core of the parent body

and stony-iron must have come from the layers between the core and the

mantle composed of silicate materials. This is possible if the bodies in the

asteroid belt could differentiate and due to collisions many of them could

be fragmented.

Large number of minerals of various types has been identified in mete-

orites. The important minerals are olivine and pyroxene. The sulphide in

chondrites is almost always FeS(troilite). In addition a variety of organics



have been identified in Murchison meteorite. Several of these are common

with biological systems

Presolar Grains

Interesting results came out of the study of anomalous isotopic compo-

sition of several elements in meteorites. These anomalies have been seen

mostly in inclusions. Here the isotopic anomalies refer to values in com-

parison to that of average solar system isotopic composition for a given

element. In particular, the noble gases are so rare, any deviation from the

average value can be quite significant.

It is hard to explain these observed isotopic anomalies in terms of the

local production mechanisms such as irradiation, mass fractionation process

or known radioactive decay. Hence, these anomalies represent materials

(grains) injected into the solar system from nucleosynthetic sites. These

are generally called as presolar grains.

Presolar grains detected in meteorites are mostly carbon bearing,

namely diamond, graphite and silicon carbide (Table 13.3). Diamond par-

ticles are more abundant than those of presolar graphite and SiC particles.

The isotopic ratio of 12C/13C determined from single SiC grains has values

around 40 to 70 (Fig. 13.2). Spherical graphite grains show variation in12C/13C ratio of around 2 to 6000 as compared to the solar value of 89

(Fig. 13.2).

The source of some of the presolar diamond particles could be super-

novae (Table 13.3). This comes from the fact that Xe isotopes 134Xe, 136Xe

and 124Xe, 126Xe present in diamonds is produced by the r -and p - processes

respectively. The presence of isotopic signature of s - process elements in

SiC grains suggest that around 99% could arise from AGB stars. The rest

of 1% SiC grains called X - grains appear to have come from supernova

ejecta. Isotopic abundances of carbon in presolar spherical graphite parti-

cles has indicated the complexity of the grain in showing that contributions

can come from various stellar sources such as Wolf-Rayet stars, supernova,

AGB (Asymptotic Giant Branch) atmospheres etc. This can be seen from

the distribution in 12C/13C ratio as shown in Fig. 13.2.

Figure 13.3 shows a section of a spherical grain from the Murchison

meteorite. It shows clearly the presence of a TiC at its centre. It is clear

that TiC must have acted as a nucleus for the graphite to condense. The

spherical graphite grain with TiC at its centre clearly indicates that it



Fig. 13.2 Shows the distribution of the isotopic ratio 12C/13C in the atmosphere ofCarbon Stars and presolar SiC and graphite grains seen in meteorites. The dashed line

refers to the solar system value of 89. The source of presolar SiC grains is consistent

with that of carbon stars. The presolar graphite grains appear to come from varioussources (Whittet, D.C.B. 2003. Dust in the Galactic Environment, Institute of Physics

Publishing, Bristol).

is presolar graphite grain. It must have originated from a carbon star.

Presolar oxide grains have also been identified in meteorites. PAHs have

been detected in samples of Allende and Murchison meteorites. Many of

the meteorites show enhanced D/H and 15N/14N ratios in bulk samples.

The primary carrier of these anomalous ratios are the organic compounds.

FeS grains in meteorites are also presolar in origin. Oxide grains are

likely to have been produced in oxygen-rich environment. Hence, pre-solar

grains formed in a distant environment could have been incorporated into

the solar nebular material with normal isotopic composition, out of which

August 4, 2010 11:23 World Scientific Book - 9in x 6in comets


Table 13.3 Presolar grains detected in meteorites.

Composition Diameter (µm) Origin

C(diamond) 0.002 SNSiC 0.3 - 20 AGB, Nova

C(graphite) 1 - 20 AGB, SNII, Nova

SiC (type X) 1 - 5 SNAl2O3(corundum) 0.5 - 3 RG,AGB

Si3N4 ∼ 1 SN II

Hoppe, P.M. and Zinner, E. 2000. J. Geophys. Res., 105,

10371: Reproduced by permission of American Geophysi-cal Union.

the planetary bodies of the solar system were formed. The volatiles of the

pre-solar inclusions could have been lost during the process of accretion

into the bodies.

Comets can in principle, explain the origin of primitive meteorites, but

not the processed mateorites which have gone through thermal evolution.

Hence fragements of asteroids may be a better source of observed mete-

orites.

The similarity between the meteorites and asteriods comes primarily

from their composition, which is reflected in their emission spectrum of

these objects. Figure 13.4 shows the emission spectrum of Trojan aster-

oid Hektor and two meteorites ALH 77003 and Tagish Lake. There is a

great similarity between Hektor and Comets Hale-Bopp and Schwassmann-

Wachmann 1. These studies indicate that C type asteroids could possibly

be the source of carbonaceous chondrites while S type asteroids may be the

source of some stony meteorites.

13.3. Meteor Streams

Another interesting aspect, presumably associated with the cometary

dust particles, is the meteor showers seen in the sky. The small particles

comprising the meteor streams revolve round the Sun, producing meteor

showers whenever their orbit crosses the Earth orbit. They are actually

made visible as they burn away in the Earth’s atmosphere. Many of the

meteor showers are seen regularly at specific times of the year (Fig. 13.5).

The most spectacular display in the sky ever recorded is that of the Leonid

meteors which were seen in the month of November of 1833. In a span of

a few hours, roughly about 20,000 meteors appear to have been seen. At



Fig. 13.3 A transmission electron micrograph of a thin section of presolar graphite grain

from Murchison meteorite. The long nucleus at its centre is in TiC crystal of 0.07 µm insize. This must have acted as a nucleation centre for graphite to condense (Bernatowicz,

T.J. 1997, In From Stardust to planetesimals, eds. Pendleton Y.J. and Tielens A.G.G.M.,ASP Conf. Ser., 122, p. 227: By kind permission of the Astronomical society of thePacific Conference services).

the present time several hundred meteor streams exist in the solar system.

What is of interest of course is the orbital characteristics of these meteors.

From the observed direction and velocity, it is possible to get the orbit of

the individual meteors in the Swarm.

It has been found that the elements of orbits of many of the meteor



Fig. 13.4 Comparison of the emissivity in the wavelength range 5 to 40 µm of two

meteorites ALH 77003 and Tagish Lake with the Trojan asteroid Hektor. There issimilarity in the curves (Emery, J.P., Cruikshank, D.P. and van Cleve, J. 2006. Icarus,

182, 496).

showers are very similar to those of the orbits of the known comets. In

1866, Schiaparelli showed that the Perseids shower had the same orbit as

the Comet Swift-Tuttle. The showers Eta Aquarids and Orionids have the

same orbit as Comet Halley and can be seen during the months of May and

October. Of course, not all the meteor showers have been identified with

the individual comets. Conversely, not all the known comets are associated

with the corresponding meteor streams. Since the association between the

two is well established only for a few cases out of a large number of known

comets (Table 13.4) and meteor streams, at first sight these results might

cast some doubt on the real association between the two. This could arise

partly due to inaccuracies in the measured parameters of the orbit of the

meteor stream. A more important reason could be that the dynamical

evolution of the meteor stream could have modified the orbit such that it is

different from that of the parent comet. Therefore a simple comparison of

the two orbits may not be appropriate. On the other hand, the position of

the perihelion of the comet and the meteor shower is a better indication of

the real association as has been shown to be true for a large number of cases.

Therefore, on a closer examination, the association between the comets and



Fig. 13.5 A typical hourly rate of meteors seen throughout the year as observed bythe Ottawa Meteor radar. The top figure represents total echo count and the bottom

curve is for echos having duration ≥ 8 secs. The peaks represent showers corresponding

to Q (Quadrantid), Y(Lyrid), E(η Aquarid), AZ (Arietid-Zeta-Perseid Complex), D(δAquarid), P (Perseid), O (Orionid), L(Leonid), G(Geminid) and U(Ursid) (McIntosh,

B.A. 1991. Comets in post-Halley Eta, eds. R.L. Newburn, Jr., Kluwer Academic

Publishers, p 557).

the meteor showers seems to be real. Let us first examine the formation

of a Swarm of particles from a comet in a qualitative manner. The dust

particles coming out of the nucleus are carried away by the gas. Since

the flow of the dust from the nucleus is maximum near about the time of

perihelion passage, it is also the place where most of the dust is introduced

into the meteor stream. Each of these dust particles becomes independent

and the orbit of these particles differs from that of a comet. The orbit of the

dust particles depends upon its velocity and the effect of radiation pressure.

This varies with the size of the particles and their physical properties. Very

small size particles are blown out of the system. The particles both lead and

lag behind the comet and eventually lead to a continuous belt within a few

revolutions of the period of the comet. The various particles in the meteor

stream are subjected to various dispersive and degenerative effects, like

gravitational perturbation by the planets, collisions and radiation pressure

and so on. Therefore the orbit of these particles evolves into orbits which

are difficult to predict. This may result in the orbit of particles in the

meteor stream quite different from that of the orbit of the parent body. In

fact the perturbations could disrupt the orbits of the particles such that it

may make them sway away for crossing the Earth’s orbit. This can possibly



explain rather poor association between the comets and the meteor showers.

But from the good correlation between the perihelion, it can be inferred

that the meteor streams are of cometary origin.

Table 13.4 Comets and their associated showers.

Comet Associated shower period ofvisibility

1P/Halley η Aquarids April 19 - May 28Orionids Oct.2- Nov.7

109P/Swift-Tuttle Perseids July 17 - Aug.24

21P/Giacobini-Zinner Draconids Oct 6 - 1055P/Tempel-Tuttle Leonids Nov 14 - 21

8P/Tuttle Ursids Dec 17 - 261861 G1 Lyrids April 16 - 25

It is interesting to see from Fig. 13.5 that some profiles have two max-

ima and others have one. This could be related to the fact that the orbital

parameters of the particles released from the comet before and after per-

ihelion passage could differ slightly, which is a function of the mass and

ejection velocity of the particles. This difference increases with time possi-

bly leading to two streams close to each other.

A very interesting study pertains to Comet Tempel-Tuttle and its asso-

ciated Leonid meteor streams covering the meteor data over the period 902

to 1969. The comet and the meteor showers have roughly the same period

of about 33 years. The comet itself does not appear to have been observed

prior to 1366, although the Leonid meteors have been recorded even upto

the year 902 A.D. From the presently known orbit of the Comet Tempel-

Tuttle, the orbit could be extrapolated backwards in time and the dates

around which the meteor showers must have taken place can be calculated.

A comparison of the calculated and the observed dates for the period be-

tween 902 to 1997 is shown in Table 13.5. The agreement between the two

suggests that most of the particles of Leonid showers are from the Comet

Tempel-Tuttle.



Table 13.5 Computed and observed dates of Leonid

Meteor Showers.

Computed Shower maximum

(Based on Tempel-Tuttle orbit) time (observed)

902 Oct 12.7 Oct 13

1035 Oct 13.2 Oct 14

1202 Oct 18.7 Oct 181366 Oct 22.4 Oct 21,22,23

1538 Oct 25.0 Oct 26

1625 Nov 7.5 Nov 4,5,61799 Nov 12.5 Nov 11,12

1866 Nov 13.7 Nov 14.11900 Nov 15.7 Nov 15-16

1932 Nov 16.5 Nov 16-17

1969 Nov 17.0 Nov 17.41997 Nov 17.4

Yeomans, D.K. 1981. Icarus 47, 492.

13.4. Particles Collected at High Altitudes

Particle collection at high altitudes of the Earth’s atmosphere is an-

other way of studying extraterrestrial dust particles and their possible ori-

gin. Various methods have been used for the collection of particles based

on recoverable rockets, balloons and aircraft. These particles are gener-

ally called Interplanetary dust particles (IDPs). The extensive collection

of IDPs from the rarefied air in the stratosphere is carried out with NASA

U2 aircraft, which flies at a height of about 20 km. The particles collected

from these flights are subjected to a thorough laboratory investigation, not

only to isolate the terrestrial contamination but also to study the morpho-

logical, structural and chemical properties of individual IDPs, in spite of

their small size. Large number of the particles can be attributed either

to rocket exhaust or to other known particle types. Many other particles

cannot be explained as contamination from known causes and they could

be extraterrestrial in origin. Roughly all these particles can be classified

into three categories; 60% as chondritic, 30% as iron-sulfur-nickel and 10%

as mafic silicate types. The chondritic particles are aggregates of small size

grains of about 1000 A. The size of the particles varies from 4 to 25µ. The

grains are highly porous as well as compact. The elemental abundances

of Fe, Mg, Si, C, S, Ca, Ni, Al, Cr, Mn and Ti in the first category of

particles are in close agreement with the bulk compositions of chondritic

meteorites. The abundances of C, S, Na and Mn indicate that the particles



are volatile rich in composition and also have a black appearance because

of the high content of carbon. The composition of iron-sulphur-nickel type

of particles is similar to that of meteoritic triolite. The silicates are olivines

and pyroxene. Evidence for the extraterrestrial origin of these dust parti-

cles comes from several sources. The direct proof for the extraterrestrial

nature comes from the observations of the presence of solar noble gases, the

deuterium enrichment (i.e. large D/H ratio) and the presence of solar flare

tracks in mineral grains within the particles. Their extraterrestrial nature

is also indicated by the presence of 10Be, which is produced by cosmic ray

bombardment.

The infrared transmission spectra between 2.5 and 25 µm of a large

number of chondrites type of IDPs shows the presence of a strong 10 µm

silicate absorption feature and possibly the 3.4 µm feature. They can be

classified into three groups as olivines, pyroxene and layer-lattice silicates.

Particles in the olivine and pyroxene group are mostly crystalline in nature

rather than amorphous. The carbonaceous material typically present is

around 2–8% of the particle by mass. Carbonates are the important sec-

ondary material in layer-lattice-silicate IDPs, due to the presence of their

characteristic feature at 6.8 µm. The Raman spectra of several IDPs show

double peaks at ∼ 1360 cm−1 and 1600 cm−1, (7.38 and 6.25 µm). These

are due to C - C vibrations of carbon. They are characteristic of aromatic

molecular units smaller than 25A. This indicates the presence of PAHs of

some kind.

The infrared spectra of two Pyrrhotite - rich grains from IDPs are shown

in Fig. 13.6. They show a strong and broad feature around 23.5 µm which is

identified with FeS stretch feature (troilite). This is similar to circumstellar

feature. Iron sulphide was also identified in the dust component of Comet

Halley during the spacecraft encounters.

The major form of non-crystalline silicate in IDPs are the GEMS (Glass

with Embedded Metal and Sulphides). The sizes of GEMS is around 0.1

to 0.5 µm in diameter. It contains nanometer sized FeNi metal and Fe -

rich sulphide grain embedded in silicate grains. The bulk composition of

GEMS are approximately chondritic for the major rock forming elements.

The chemical and mineralogical information indicate that pyroxene-rich

IDPs and olivine IDPs are from comets (Fig. 13.7). The overall properties

pertaining to the physical and chemical nature of these particles strongly

suggest that many of them are of cometary origin containing possibly some

component of presolar origin. Supporting evidence comes from the presence

of meteor showers as well as the factual information that the vapourized



Fig. 13.6 Infrared Spectra of IDPs (U2012A-2J and L 2011∗B6), Herbig Ae/Be stars

(AB Aur and HD 163296) and FeS (troilite). The residual spectra for AB Aur andHD 163296 is shown after subtracting the best model fit from the observed spectra. The

dust components used in the model are, glassy silicate, forsterite, carbonaceous material,metallic iron and water ice. The mineral of U2012A-2J is dominated by pyrrhotite and

that of L2011∗B6 by low-NiFi-Sulphides (Keller, L.P., Hony, S., Bradley, J.P. et al. 2002.Nature, 417, 148).

material of the comet at each apparition is dispersed into the interplanetary

medium, some of which may find its way on to the Earth.

FeS grains seen in IDPs, meteorites, comets, young and old stars indicate

the similarity among them.



Fig. 13.7 Histograms comparing the submicrometre-scale distribution of Fe/(Fe + Mg)

in IDPs with Comet Halley dust. The horizontal scale range from 0 to 1. (a) Chon-dritic porous pyroxene IDP U222B28 which contains abundant Mg-rich silicates, mostly

enstatite (Mg SiO3) and some forsterite (Mg SiO4) result in large Fe/(Fe +Mg) data

points near zero. (b) Halley(PUMA) mass spectrometry data (Bradley, J.P., Snow, T.P.,Brownlee, D.E. and Hanner, M.S. 1999. In Solid Interstellar Matter: The ISO Revolu-

tion, eds. L. d’Hendecourt, C. Joblin and A. Jones, Springer-Verlag, p. 297: with kind

permission of Springer Science and Business Media).

13.5. Primordial Material

The material out of which the Sun and the planets were formed does not

refer to the original material present at the time of formation of the universe

about 13× 109 years ago, but rather to the modified material which existed

around 4.5×109 years ago. During all these times, the interstellar material

was being constantly enriched with heavier elements through the element

building in stars followed by ejection of this material into the interstellar



medium. The same process has been happening since the formation of the

solar system to the present time. It is not clear at present at what time in

the history of the universe the Oort cloud was formed and hence its chem-

ical composition is also not known. This will in turn be reflected on the

original chemical composition of comets. For example, if the comets were

formed along with other solar system bodies about 4.5×109 years ago, they

would have the same composition as that of the solar system material. But

on the other hand, if they were formed more recently they would have a

different composition reflecting the contemporary interstellar abundances.

One method to get information about the possible nature of the primordial

cometary material and hence the time scale or the age is through the study

of the isotopic ratios of various elements. As is well known, the relative

abundances of various isotopes preserve the life history of the formation

process and hence help in understanding the nature of the original mate-

rial. The isotopic ratio of several elements has been determined for comets

(Table 5.4). The measured isotopic ratios are 12C/13C ∼ 90, 14N/15N ∼300, 32S/34S ∼ 22 and 16O/18O ∼ 450 and is similar to terrestrial values

of 89, 270, 24 and 500 respectively. Therefore the nature of the material at

the time of formation of the Oort cloud is similar to that of the primordial

cometary material.

13.6. Chemical Evolution

The origin of life on Earth has intrigued mankind since early times. In

the standard scenario, the production of organics is the starting point of the

whole complex process, as all life on Earth is composed of organic material.

It is remarkable that relatively only a smaller number of organics appears

to have been used in forming life system among a variety of organics that is

possible. The compounds of major interest are those normally associated

with water and organic chemistry in which carbon is bonded to itself and

to other biogenic elements. The biogenic elements, H, C, N, O, S and P,

are generally believed to be essential for all living systems. In 1953, Miller

showed that when gaseous mixture of NH3, CH4 and H2O is subjected

to an electrical discharge, it produced various kinds of organic molecules

including amino acids. This remarkable experiment showed for the first

time the possibility of synthesis of organic molecules from a mixture of

simple gases in the presence of an energy source. This led to the suggestion

that a similar type of process could have taken place in the early stages of



the earth leading to organics, the basic ingredient for the formation of life.

However, studies have indicated that these simple molecules are unlikely

to be present as major constituents in the early atmosphere as they are

most probably photolyzed in a time scale of year or so. In addition, other

studies indicate that the early Earth’s atmosphere contained mostly CO2,

H2O and N2, which makes it difficult for the formation of organics. The

existing observations show that the amount of organics in the solar system

objects seems to increase with distance from the Sun. Therefore, it is

intriguing that the organics necessary for chemical evolution is found in

the outer solar system, whereas water, an essential ingredient for the life

formation, is found in the inner solar system. This led to the suggestion

that the organics might have been transferred from outer to inner regions

of the solar system by some means, possibly through comets. Therefore

the general conclusion is that it is highly unlikely that the biogenic could

have formed on the Earth itself and therefore it had to be transported to

the Earth through some means. The early chemical evolution was then

believed to be followed by biological evolution, finally leading to life on the

Earth. Even though this scenario is being taken as a working model, it is

not at all known at the present time, how the whole process can take place

in such a manner.

In recent years, the possible role of comets during the early chemical

evolution on the Earth has been put forward. This is basically related to

the fact that molecules of various complexities have been seen in comets.

In particular, observations of Comet Halley showed for the first time the

presence of organics in comets through the detection of CHON particles.

The composition of these grains showed that they are made up of highly

complex organic molecules of various kinds. Comet Halley observations

indicated the possible presence of Phosphorus in the mineral core of the

particles. Therefore, all the biogenic elements (H, C, N, O, P and S), es-

sential for living system, have been detected in comets. The direct evidence

for the existence of building blocks of life in comets came from the detection

of Glycine (an amino acid) for the first time in Comet Wild 2. There are

several ways in which the organic material could have been incorporated

into comets. It could have been incorporated directly at the time of for-

mation from the material in the solar nebula which is a typical interstellar

cloud. More than 150 molecules have been detected in interstellar clouds,

most of which are organic in character. Therefore, organics in comets could

be interstellar in origin.

There are several observations to show that the earth does seem to



receive cometary material, such as meteor showers, dust particles collected

at high altitudes, sedimentation at deep sea, cratering rates and so on.

Geological evidences also indicate that comet collision with earth must have

been high during earlier times. Therefore, the influx of cometary material

could have injected large amounts of simple and complex organic molecules

on the Earth. Particularly, the molecules accreted during the early times

of planet formation when the comet collision was high could have triggered

the chemical evolution cycle on the Earth. However, it is not clear whether

these complex molecules could survive during the time of impact. This

depends to some extent on the dynamics of the process.

At the present time there is evidence for the presence of some form of

life on the Earth on or about 3.5× 109 yrs ago. For times earlier than this,

it is anyone’s guess. Is it then possible that the organic molecules pouring

from comets on to the Earth, particularly during the early times, might

have started the whole cycle of chemical evolution, finally leading to the

origin of life on the earth?

It is interesting to note that the presence of Glycine in comets and its

delivery to objects indicate that the building block of life is pervasive in

the universe which support the hypothesis that life is a cosmic phenomena.

In fact earth is most likely place as it had liquid water, proper oxidation

- reduction ratio and right chemistry to get the whole process started. This

suggestion has been further extended to the possibility that some sort of ba-

sic primitive cells themselves could have come from the cometary material

on to the Earth. It is interesting to see the similarity of the relative abun-

dances in comets and in living organisms as shown in Table 13.6. In this

connection it is also interesting to see the extreme conditions under which

microorganisms can exist. It is indeed remarkable that many microorgan-

isms have been found to be able to survive under extreme environmental

conditions, such as boiling water, freezing water, irradiation etc. They seem

to adapt themselves to these environments and they are able to survive and

grow. But one basic limitation is that they require liquid water. It has been

suggested that 26Al present in the early stages of solar system, which has a

half life less than 0.7 million years, would have decayed within a few million

years of formation of the solar system giving heat that could have melted

the cometary material. Therefore it appears that comets could have played

an important role in the process of chemical evolution, which finally led to

life on Earth. However, many of the evidences brought forth to arrive at

this conclusion have not yet been firmly established.



Table 13.6 Relative abundances in life and in

cometsa.

Element Bacteria Mammals Comets

Hydrogen 63.1 61.0 56Oxygen 29.0 26.0 31

Carbon 6.4 10.5 10

Nitrogen 1.4 2.4 2.7Sulphur 0.06 0.13 0.3

Phosphorous 0.12 0.13 0.08b

Calcium - 0.23 -

a. In %

b. (0.08) is the cosmic abundance.Delsemme, A.H. 2000. Icarus, 146, 313.

13.7. Terrestrial Water

It is interesting that around 70% of the Earth’s surface is covered with

water (sea water, oceans). The origin of this terrestrial water has been the

subject of investigation for a long time. There is no general agreement on

this issue at the present time. It might have come from the earth itself or

it could have come from an external source such as comets.

As mentioned earlier, water could have been released by the earth itself.

This refers to the accumulation of primordial water from the solar nebula

adsorbed on to the grains when it was formed from the accreting disk. The

amount of water accreted by the earth may be around 1–3 earth oceans

of water. However the accretion process leading to the formation of the

earth passed through several violent events, during which it could have also

been melted in the final stages. It is not clear as to what fraction of the

accumulated water was retained by the earth during the accretion process.

The other suggestion is that the ocean water on the earth could have

come from the accumulation of the material from comets when the impact

rate was much significant in the past during the formative stages of the

earth. The D/H ratio of Jupiter and Saturn most probably representing

the material referring to solar nebula gas has a value 2.1 × 10−5. This is

consistent with D/H ratio derived from the solar wind implanted into lunar

soil. The D/H ratio derived from three Comets Halley (3.2 × 10 −4), Hale-

Bopp (3.3 × 10−4 and Hyakutake (2.9 × 10−4) ∼ 3 × 10−4. Therefore the

D/H ratio of these comets is about two times that of terrestrial water D/H



ratio (1.5 × 10−4). Several laboratory studies have shown that D/H ratio

in sublimating ice can increase or decrease with time depending upon the

fractionation process (such as ion-molecule reaction etc) and on the nature

of the material. It is possible that comets which contributed water to earth

may be different from the comets for which D/H ratio has been measured.

13.8. Impact of Outside Bodies

The impact of outside bodies in the solar system is a natural phenomena.

This becomes obvious from the craters seen on the surface of planets and

their satellites, including the earth. The minor bodies which impinge on

the objects in the solar system are the comets and asteroids. But in recent

years the distinction betwen comets and asteroids is becoming nebulous as

many asteroids are extinct comets or dormant comets. There is also a group

of short period comets which come close to the earth. The impacting body

possess tremendous amount of energy and can cause devastigating effect

on the object being collided. In this connection it is interesting to know

whether there is any evidence for such a process to have taken place on the

earth’s surface. The impact could have moderate or catastrophic effects on

the geophysical, biological or climatic changes on the earth. They could

possibly appear in terrestrial records.

There are several craters present on the earth’s surface like, crater in

Arizona, U.S.A. which is approximately 1.2 km in diameter and 100 meter

deep. This crater is believed to be due to the impact of a iron meteoroid of

size 30 metres moving with a speed of about 20 km/sec about 50,000 years

ago.

The Tunguska event of 30 June 1908 in Central Siberia in which all the

forest trees near Tunguska river were flattened upto about 30 to 40 km was

by a shock wave. This shock wave was created by the incoming object which

did not survive to hit the ground but exploded at an altitude of about 8

km. For many years, the cause of this event was considered to be a comet.

However it is likely that stony-iron meteorite must have been responsible

for this event as minute fragments of iron -nickel were found in the areas.

The diameter of this stony-iron material could be around 60 meters.

Many meteorites survive through the earth’s atmosphere and fall on the

earth’s surface. They are generally identified due to their characteristic ap-

pearance. There are some meteorites which are destroyed before they reach



the earth’s surface. Two well known meteorites of this type are the Murchi-

son and Allende meteorites. Before they could reach the earth’s surface,

they exploded and shattered into pieces whose remnants were recovered.

Both the meteorites are carbonaceous chondrites.

As mentioned already, fossil records suggest the presence of biological

extinction event repeating on the earth with a period of around 26 my. It is

suggested that cometary showers impinging on the earth could have caused

such phenomena.

The credibility for the hypothesis that outside bodies impinge on the

earth came when 20 pieces of Comet shoemaker-Levy 9 pierced through

Jupiter’s atmosphere during the period from July 17 to 22, 1994. This

is indeed an historical event that was witnessed from earth. Therefore it

is possible that such an event can happen on the earth in the future and

hence poses danger to earth. Therefore efforts are made to see how to

prevent such an event taking place on the earth. For this purpose the sky

is being monitored continuously to look for objects which might come close

to the earth, catalogue them and keep track of these objects. Some of them

are Near-Earth Object Survey, the Near-Earth Asteroid Tracking. The

Near-Earth objects (NEOs) are defined as asteroids and comets which have

perihelion distances of 1.3 AU or less. The total number of NEOs/NEAs is

estimated to be around 700–1300.

The only way to avoid the collison is to destroy the object or change

its track. Several suggestions have been put forward which can achieve

these objectives. One suggestion is to explode the object. But one problem

with this approach is that it would shatter the body into pieces, some of

which might still head towards the earth. This is blast-and-hope strategy

i.e. nothing comes towards the earth. Another way is to land a spacecraft

on the surface of the object in order to gently push it off the course. This

involves a rather complicated manovering. Another suggestion is to push a

heavy object near the incoming object for long enough time that it could

produce sufficient gravitational tug to change its orbit. Similarly, solar

radiation pressure beaming on the object may be used to alter the orbit of

the incoming object.

All these suggestions require a spacecraft to be put out there at the

right time, but it requires lot of advance planning. Hopefully with man’s

inguenity backed by sophisticated technology, it is possible to come out

with a simpler way of tackling this problem.



13.9. Overview

The nature of the original material of the solar nebula (a typical inter-

stellar cloud) and its relation to comets and other objects in the solar system

and their inter-relationship is of great interest. This is shown schematically

in Fig. 13.8.

Interstellar Matter (Gas and Dust)

Meteorites

Condensation Process

Stars (Element Synthesis)

Solar Nebula

Asteroids

Comets

IDPs Meteor Streams

Explosion Process (Nova and Supernova)

Fig. 13.8 A Schematic representation showing the comets formed out of a typical in-terstellar cloud (solar nebula) containing presolar grains emitted from stars, novae and

supernovae and its relation to other objects in the solar system.

The observed silicate dust mineralogy is quite similar among IDPs, me-

teorites, asteroids, comets and in circumstellar shell of stars. The high value

of the isotopic ratio of several elements observed in some of these objects

as compared to solar system value indicate the presence of presolar grains.

In particular, the presolar grains with high value of 12C/13C ratio, con-

tribution must have came from various stellar sources such as Wolf-Rayet

stars, novae, supernovae, AGB atmospheres etc. The source of IDPs rich

in deuterium is attributed to molecular clouds. Similarly FeS grains have

been seen in IDPs, meteorites, comets and from young and old stars. The

presence of FeS grains in all these kinds of objects give a direct link between



the solar system material (comets) to the presolar material. Therefore it is

likely that sulphur is tied up in grains as FeS. Hence it is likely, that the

solar system bodies formed out of the solar nebular material had already

crystalline FeS grains.

Carbon chemistry or organics may be used as a probe for a qualitative

discussion of this cosmic connection. Before coming to a discussion of this

aspect, it may be worthwhile recapitulating some of the points regarding

organics seen in various objects in the solar system and in interstellar space.

Evidences for the presence of organic molecules in interstellar clouds

came from the observations made in the radio frequency region. A com-

pilation of some of these molecules is given in Table 13.7. A close look at

the table reinforces the belief of the presence of numerous different organic

compounds in interstellar clouds. The exact nature of the organic compo-

nent of the dust is also not known. But it could be Polycyclic Aromatic

Hydrocarbon (PAHs).

The characteristic feature of the solar system is that the rocky planets

(Mercury to Mars) form the inner region and icy planets form the outer

regions (Jupiter to Neptune) and are separated by asteroids. The reduc-

ing atmosphere of the outer planets consisting of H, He, CH4 and NH3 is

conducive to the formation of organic molecules. In fact several carbon

compounds have been detected on Jupiter. Complex organic chemistry is

operating on Titan, the Satellite of Saturn. The Earth has also organic

material as the life on Earth goes back to about 3.5× 109 years.

The new class of grains present in comets, called CHON particles, es-

tablished that a large fraction of the comet is made up of very complex

organics. They are shown in Table 13.8. They provide evidence for the

existence of many classes of cyclic and acyclic organic compounds. They

include unsaturated hydrocarbons such as Pentyne, hexyne etc, nitrogen

derivatives such as hydrogenic acid, aminoethylene etc., Heterocyclics with

nitrogen such as purine and pyridine etc. HCN and H2CO volatiles, allem-

ine and pyrimidines and their derivatives which are biologically more sig-

nificant has been seen in comets. The possible presence of Formaldehyde

Polymers (H2CO)n has also been suggested. It is quite possible that CHON

particles contain many more molecules than identified so far due to the fact

that the impact velocity of the dust at the dust mass spectrometer was very

high (∼ 70 km/sec), which could possibly have destroyed many of them.

The carbon content in cometary dust is close to 25% by weight. The or-

ganics present in the dust particles of Comets Wild 2 and Tempel 1 have

been studied extensively. The collected samples from Wild 2 has been put



Table 13.7 Some of the observed interstellar molecules.

May 18, 2010 19:31 World Scientific Book - 9in x 6in comets


Table 13.7 Some of the observed interstellar molecules.

(a) Inorganic species (stable)

Diatomic Tri-atomic 4-atom 5-atom

H2a HCI ? H2Oa NH3a SiH4b

COaPN H2Sa

CSaNaCIb SO2a

NO AICIb OCSNS KCIb

SiOa AIFb

SiSa

(b) Organic molecules (stable)

Alcohols Aldehydes and ketones Acids Hydrocarbons

CH3OH methanol H2CO formaldehyde HCNa hydrocyanic C2H2b acetylene

EtOH ethanol CH3CHO acetaldehyde HCOOH formic C2H4b ethylene

H2CCO ketene HNCO isocyanic CH4 methane(CH3)2CO? acetone

Amides Esters and ethers Organo-sulfur

NH2CHO formamide CH3OCHO methy formate H2CS thioformaldehydeNH2CN cyanamide (CH3)2O dimethy ether HNCS isothiocyanic acid

NH2CH3 melthylamine CH3SH methyl mercaptan

Parafin derivatives Acetylene derivatives Other

CH3CNc methyl cyanide HC3Na cyanoacetylene CH2NH methylenimineEtCN ethyl cyanide CH3C2H methylacetylene CH2CHCNa vinyl cyanide

(c) Unstable molecules

Radicals Ions Rings Carbon Isomerschains

CH C3Ha,c CH+ SiC2b C3Sa HNCa

CNa C3Na HCO+ C3H2a HC5Na CH3NC

OHa C3Na N2H+ Ca3 HC7Na

SOa C4Ha HOCO+ HC9Na

HCO C5H HCS+ HC11Na

C2b C6H H3O+? CH3C3NC2Ha C2 Sa HCNH+ CH3C4H

C2b CH2CN H2D+? CH3C5N?

(Turner, B.E. 1989, Space Sci. Rev., 51, 235).

a Seen in CSE as well as ISM sources.b Seen only in CSEs.

c Both linear and cyclic forms.

to intensive laboratory investigations. The results of these investigations

corroborate Comet Halley results. All these studies have given startling

results in showing the presence of wide range of organics of various types,

the existence of which was hard to imagine. The overall character of the

dust in comets i.e. silicate and carbon, is similar to that of interstellar

grains.

The mineralogical composition of meteorites reflect differing degrees of

thermal evolution indicated by Stony-iron and iron meteorites. Carbona-



Table 13.8 Types of organic molecules in Comet Halley dust.



ceous chondrites seems to be primitive in the sense that they have not al-

tered much their state from the time of formation. The organic material

found in carbonaceous chondrites is highly complex. In Murchison mete-

orite, different amino acids have been identified. Many of them occur on

earth biologically. A few of the different compounds seen are the following:

Carboxylic acids, aliphatic and aromatic hydrocarbons, amines and amides,

alcohols, ketones, purines, pyrimidine etc. The organic matter found in

meteorites probably existed in the solar system a billion years before the

appearance of life on the Earth. There are many similarities between the

organic compounds in dust in comets and chondrites. The carbon content

in carbonaceous chondrites is ∼ 3 to 5%, which is low by almost an order

of magnitude compared to cometary dust. There is also an indication that

they must have passed through a high temperature phase as well (∼ 400 K)

in order to explain some of the compounds present in them. Therefore it is

unlikely that comets are the source of meteorites as they remained at low

temperature. Hence asteroids is most likely the source of meteorites. Ob-

servations show similarities between asteroids and meteorites. The isotopic

anomaly seen in chondrules in meteorites is attributed to outside material

survived and accreted and are therefore pre-solar grains.

Many of the IDPs which are thought to be of cometary origin is car-

bonaceous in character.

The various objects in the solar system are generally believed to have

been formed as a by product of the formation of the Sun. The Sun was

formed out of a typical interstellar cloud referred to as the primitive solar

nebula. Model studies have shown that when contraction of the cloud takes

place, the material will form a disk around a central core due to conserva-

tion of angular momentum. This material composed of gas and dust will

accumulate in increasing numbers and form a thin disk of particles. This

will then become unstable against gravitational forces and separate out

from the system. These rings then clump together and form planetesimals

and move in circular orbits around the centre of the nebula. These plan-

etesimals act as nuclei for the formation of planets. This is also responsible

for the nearly coplanar arrangement of planets and for a common direction

of revolution of the planets around the Sun.

Condensation model calculations have shown that different types of ma-

terial condense out of different temperature and densities, which imply dif-

ferent distances from the Sun. So there could be materials of high temper-

ature phase, low temperature phase as well as a mixture of the two in the

solar system objects. Carbon compounds can be vapourized upto around 2



to 3 AU and water upto around 5 AU. Therefore terrestrial planet zones will

be rocky and the outer planet zones will be made up of water, methane, am-

monia and CO2-ice. This is in rough accordance with the gross properties

of objects seen in the solar system. However, the detailed understanding

of the whole problem is far from clear. The exact nature of compound

produced depend upon various factors such as temperature, density, chem-

ical composition of the constituents and various physical processes such as

heating, cooling, annealing, selective evaporation, fractionation and so on.

It is found from laboratory experiments that all the compounds seen in car-

bonaceous chondrites can be produced from CO, H2 and NH3 on a Fe2O3

or clay catalyst at a temperature of around 400 to 430 K. Hence it is not

surprising that there is a systematic change in composition going outwards

from the Sun. Comets are produced in a cold temperature phase in the

outskirt of the Solar System and so it has kept most of the molecules it had

from the original material.

During the time interval of the formation of the Solar System, materials

could have been injected into the solar system from outside (like from a

supernova explosion). Some of this material may be preserved in the solar

system objects as inclusions etc. In addition, comets colliding with Earth

during the early stages of the formation could have deposited organic mate-

rial, which might have started the chemical evolution on the Earth. Comets

could also account for the biosphere on the Earth.

Therefore the organic material present in interstellar clouds and in solar

system objects are the same material to start with but may look different in

some of the solar system objects at the present time due to variable thermal

evolution in these objects depending on their radial distance from the Sun

(Cosmic Evolution).

So far we have been discussing the possible interrelationship between

various solar system objects and comets. In certain cases the relationship

to comets is more direct than in others. This is partly due to lack of good

data as well as to the inadequacy in our present understanding of the nature

of these objects. With further studies, it is hoped to understand the origin

of the solar system itself and possibly even the origin of life on the Earth.

Problems

1. In the study of the origin of life, one deals mainly with the elements

H,C, N and O. Explain.

2. Is there evidence of life beyond the Earth? If the answer is negative,



discuss why?

3. Explain how the number of planetary systems in the Galaxy, with life,

like on the Earth can be estimated

4. The meteor streams seen in the sky appear to diverge from a point in

the sky. Explain.

5. What is the range of velocities at which meteors encounter the Earth?

What is the explanation for the limiting values?

6. What has Titus-Bodes law to do with the question of asteroid belt?

7. Meteorites are said to act as beacons to the past. Explain.

References

The following reference gives a good account of asteroids

1. Bottke, W.F.Jr., Cellino A., Paolicchi, P. and Binel, R.P. (eds). 2003.

Asteroids III, Univ. Arizona Press, Tucson.

The following reference gives a good description of meteorites

2. Lauretta, D.S. and McSween, H.Y. Jr. (eds.) 2006. Meteorites and the

Early Solar System II, Univ. Arizona Press, Tucson.

For Presolar grains reference may be made to

3. Lugaro, M. 1995. Stardust from Meteorites, World Scientific, Singa-

pore.

The technique of particle collection at high altitudes and the results of such

studies are given in the following review articles.

4. Brownlee, D.E. 1978. In Cosmic Dust. ed. McDonnell, J.A.M. New

York: John Wiley and Sons. p. 295.

5. Sandford. S.A. 1987. Fund. Cosmic Phys. 12, 1.

The early work on the various processes leading to element building in stars

is discussed in the classic paper:

6. Burbidge, E.M., Burbidge, G.R., Fowler, W.A. and Hoyle, F. 1957.

Rev. Mod. Phys. 29, 547.

For more discussion reference may be made to

7. Prantzos, N., Vangioni-Flam, E. and Casse, M. (eds.) 1993. Origin and

Evolution of the Elements, Cambridge: Cambridge University Press.

The Oparin-Haldane hypothesis of the Origin of Life can be found in

8. Ponnamperuma, C. 1982. Extraterrestrials: Where are they? eds.

Hart. M.H. and Zuckerman, B. New York: Pergamon Press, p. 87.

The other view point of cometary origin can be found in the following ref-

erences.



9. Hoyle, F. and Wickramasinghe, N.C. 1981. Comets and the Origin of

Life ed. Ponnamperuma, C. Dordrecht: D. Reidel Publishing Company,

p. 227.

10. Wickramasinghe, N.C. 2009. Astrobiology, World Scientific Publishers,

Singapore.

The following book may be referred for solar system studies.

11. Lewis, J.S. 1995. Physics and Chemistry of the Solar System, Aca-

demic Press.

The following papers pertains to Terrestrial water, Impact of outside bod-

ies, asteroid-comet connection and binaries.

12. Drake, M.J. and Campins, H. 2006. In Asteroids, Comets and Meteors,

eds., D. Lazzaro, S. Ferraz-Mello and J.A. Fernandez, Cambridge Univ.

Press, Cambridge, P.381.

13. Morrison, D. et al. 2003. In Asteroids III, eds. W. Bottke, A. Cellino,

P. Paolicchi and R. Binzel, Univ. Arizona Press, Tucson.

14. Toth, I, 2006. In Asteroids, Comets and Meteors, eds. D. Lazzaro, S.

Ferraz-Mello and J.A. Fernandez, Cambridge Univ. Press, Cambridge,

p. 67.

15. Noll, K.S. 2006. In Asteroids, Comets and Meteors, eds. D. Lazzaro, S.

Ferraz-Mello and J.A. Fernandez, Cambridge Univ. Press, Cambridge,

p. 301.


CHAPTER 14

Problems and Prospects

14.1. Epilogue

So far we have discussed in some detail our present understanding with

regard to various cometary phenomena which is summarized in Fig. 14.1

We have a reasonable knowledge of the origin, nature and composition of

comets. Though much progress has been achieved in the last ten years since

the publication of the second edition, many of the aspects are still not well

understood.

Over the years, our knowledge with regard to comets has improved

enormously from the studies based on several bright Comets like Ikeya-

Seki, Kohoutek, West, Bradfield, Hale-Bopp, Hyakutake and others. The

Comet IRAS-Araki-Alcock, which came to within about 4–7 million kilome-

ters from the Earth in May 1983, gave the opportunity to observe a comet

at such a close approach. Comet Kohoutek gave a big boost to cometary

science in 1974 through a concentrated and co-operative effort made by the

scientists working in various fields. This venture actually created an inter-

est among the physicists, chemists and biologists as well, in problems con-

nected with cometary science. It also created an increased interest among

the astronomers who normally work in other fields. At present we have

beautiful techniques for both the ground based and space observations in

different spectral regions, remarkable new instrumentation, (ground based

equipment and computers), which are much better than what was available

about 30 to 40 years ago. With these advances, phenomenal progress in

the understanding of comets has been achieved.

However, there are still questions which cannot be answered, either

from ground based or satellite observations. The only way to understand

these problems is through probes or missions to comets which can pass

435



Fig. 14.1 The various processes arising out of the interaction of solar radiation and

solar wind with a comet, which result in the observed features in a comet are shown in ablock diagram (Report of the Science Working Group, The International Halley Watch,

July 1980).

close to the nucleus. This was achieved with remarkable success for the

first time during the apparition of Comet Halley in 1986. A target for

space missions has to satisfy several requirements. First, the comet’s orbit

should be predicted accurately. This means that the arrival time of the

comet can be predicted well in advance. This is very essential because of

large time periods that are involved for developing experimental packages

and the necessary preparations for making space flights. The comet should

be bright, as well as exhibit as far as possible all the observed phenomena

for the maximum scientific return. The condition that the orbit should be

well known mean that the comet should have returned at least a few times


Problems and Prospects 437

and this eliminates new comets. It leaves only the short-period and the

intermediate-period comets as potential candidates. Comet Halley satisfied

all the requirements for a space mission and is also the most famous of the

known comets, which has been observed for the last 2000 years (Table 14.1).

Table 14.1 Perihelion passages of Comet Halley.

240 B.C. May 25 (probably observed) 912 Jul 19

164 Oct 13 (not observed) 989 Sep 687 Aug 6 1066 Mar 21

12 Oct 11 1145 Apr 19

66 A.D. Jan 26 1222 Sep 29141 Mar 22 1301 Oct 26

218 May 18 1378 Nov 11

295 Apr 20 1456 Jun 10374 Feb 16 1531 Aug 26

451 Jun 28 1607 Oct 27

530 Sep 27 1682 Sep 15607 Mar 15 1759 Mar 13

684 Oct 3 1835 Nov 16760 May 21 1910 Apr 20

837 Feb 28 1986 Feb 10

ESA Giotto mission pamphlet, 1981 (upto 1910)

The in situ studies of Comet Halley have been followed by Comets

Giacobini-Zinner, Grigg-Skjerllerup, Borrelly, Tempel 1 and Wild 2 (Ta-

ble 1.3). In addition, the cometary material brought back to earth from

Comet Wild 2 gave the first opportunity to study the true nature of the

cometary material. All these studies gave a large number of unexpected

results as well as showed the complexity of the physical processes occurring

in the coma. These observations, combined with ground based and satel-

lite observations, covering the entire range of the electromagnetic spectrum

from X-rays to radio wavelengths provided the complete set of data on sev-

eral comets. These results have increased our knowledge about cometary

science in a dramatic way.

14.2. Future studies

Although space missions to comets provided a remarkable insight into

our understanding of the cometary phenomena, still there are several fun-



damental questions, which remain to be answered with regard to the origin

and evolution of comets, structure and composition of the nucleus, par-

ent molecules, the interaction of solar wind with cometary plasma, relation

to chemical and biological evolution and so on. In addition, the space

missions to comets provided only snap shots of the physical and chemical

conditions in these comets, as the observations could be carried out only for

a very limited time period of a few hundred to a few thousand seconds or

so. Hence one should have a cautious approach in extending these data to

other comets and in deriving general conclusions. Much more data on other

comets, similar to that achieved for several comets, is needed before arriv-

ing at general conclusions regarding the similarities and differences among

comets of different types. It will also be of interest to probe the nucleus

of a long period comet, coming directly from the Oort cloud, in order to

examine the possible presence of the aging effect among comets.

The next logical step would be to achieve long duration close encounter

of a short period comet to study the time evolution of the development

of the coma as a function of the heliocentric distance. This can provide

information about the nucleus, the elemental and molecular composition of

the gas and the dust, solar wind interaction and so on. These studies will

go a long way in comparison to those of flybys. This can be achieved by

putting a satellite with instruments around a comet and tracking it as it

moves with the comet. It should also probe the nucleus with a penetrator

for physical and chemical studies. This is the objective of Rosetta mission

which will rendezvous with Comet 67P/Churyumov-Gerasimenko around

2014. It will also have a Lander for making on-spot studies. The Rosetta

mission with planned capabilities is an ambitious program.

A new generation of powerful ground based optical/IR/Radio telescopes

of large collecting area will be available in the near future. Combining this

feature with advanced instrumentation and state of the art in optical and

IR detectors will provide excellent opportunities for making very high res-

olution imaging, spectroscopy, polarimetry and so on. The extraordinary

sensitivity combined with high spatial resolution should give exciting re-

sults. It will also be possible to extend the observations to faint objects.

These will be supplemented with high quality observations carried out with

satellites in the UV and X-ray regions. With these instruments, it should

be possible to extend the observations to faint objects and also look for

comet-like objects in the Oort cloud. There is, thus, a bright future for

cometary observations, with both the ground based and above the atmo-

sphere instruments.


Problems and Prospects 439

The laboratory simulation studies of the processes that could occur

in a cometary environment is another important input necessary for an

understanding the behaviour of comets. Of course, it is very difficult to

create the actual cometary environment in the laboratory. In addition,

it is very difficult to simulate a cometary target as we do not know the

exact compositions and the physical state of a comet. However, studies

carried out on approximating the conditions of real comets will help for the

development of more realistic comet models.

Future studies, based on combined effort of space missions, ground based

and above the atmosphere observations, laboratory investigations and the-

oretical studies should give new insights into the structure, evolution and

origin of comets. Hopefully, through a series of such studies, it will be pos-

sible to resolve many of the basic and fundamental problems which have

no answers at the present time. These studies should also give enormous

information with regard to the early history of the solar system; physico-

chemical, dynamical and thermodynamic conditions existing at that time;

the information and relation to other solar system objects; the relation to

interstellar molecules, chemical evolution, the origin of life and Exobiology,

space plasma physics and so on. Therefore cometary science is an exciting

area of study for years to come.


Index

aberration angle, 305–307absorption, 227, 228, 233, 235, 240abundances, 95, 129, 131, 132, 137,

138, 176, 196, 197, 203, 206, 208,345, 359, 360, 392, 393

abundances of Heavy Elements, 128acceleration, 303, 321–323, 325age, 356, 358albedo, 227, 230, 231, 235, 255–257,

260, 299, 340, 341, 344, 350–352,355, 377

annealing, 68, 69anti-tail, 18, 23, 220, 221, 225, 268,

269, 276, 277appearance, 1–3asteroids, 401, 402Atlas of Cometary Forms, 323, 336Atlas of Representative Cometary

Spectra, 93atomic spectroscopy, 51

band polarization, 123band sequence, 55, 57Bayeux tapestry, 2bend in the tail, 326binary systems, 379biological extinction, 388black body of temperature, 70black body radiation, 49blocking coefficients, 99Boltzmann Distribution, 64Borrelly, 22, 25, 29, 303, 355

bow shock, 310, 313, 314, 319, 320,322

brightness, 10, 12–15, 30brightness profile, 144, 205, 254Bruggeman rules, 248

C2 molecule, 113C-H stretch, 85, 288chemical composition, 279, 283, 290chemical diversity, 206chemical evolution, 421, 422, 431chemical subgroups, 57CHON particles, 28, 90, 167, 201, 290circular polarization, 232, 266clathrate hydrates, 344collisional excitation, 106, 107, 122,

172column densities, 162, 169, 172, 176,

178, 181, 182, 188, 193coma, 88–90, 92, 105, 107, 115, 118,

120, 135, 138cometary fading, 384cometary showers, 387complex molecules, 86, 90, 144, 174,

191, 202, 209composition, 129, 132, 162, 165, 175,

206, 339, 340, 347, 348, 359, 360,392, 396, 397, 401, 403, 407–411

Condon parabola, 60continuum, 249–253, 255, 256, 259,

261, 274, 297, 299Cretaceous-Tertiary boundary, 388

441



Deep Impact, 22, 31, 284, 290

density of grains, 258

diatomic molecules, 58

dielectric constant, 241, 243–246

disconnection events, 328

discovery, 1, 4, 6, 16, 28

Discrete-Dipole approximation, 242,247

Dissociative Equilibrium, 49

Doppler shift, 50

dust features, 223

dust particles, 249, 250, 254, 255, 257,259, 261, 264, 267–270, 281, 282,285, 287, 290, 292–294, 296, 298

dust production, 151, 259, 260, 272

dust tail, 17, 18, 22, 27, 217, 220, 222,225, 226

dust trails, 222

dust-to-gas ratio, 260

dynamical aberration, 305, 336

dynamics, 213, 220, 226

efficiency factors for scattering, 233

electronic transition moment, 62, 63,104, 112

electronic transitions, 54, 56, 59, 61,74

elemental abundances, 129, 130, 182

emission lines, 57, 60, 66, 70–72, 74

energy density, 48, 49

energy level diagram, 96, 98, 109, 121

equation of state, 49

equilibrium constant, 50

evolution of orbits, 393

excitation temperature, 110, 125, 128,129

expanding halos, 356

extended source, 167

extended source of molecules, 175

extraterrestrial dust particles, 416

Fabry-Perot, 91, 94, 177, 190

filamentary structure, 324

flares, 15, 16, 348

fluorescence efficiency factor, 123, 142

fluorescence process, 95, 102, 105,118, 121, 122

flyby missions, 27forbidden lines, 90, 92, 93Forbidden Transitions, 90, 121Fountain model, 184, 187, 254Franck-Condon factors, 60, 62, 63Fraunhofer lines, 96, 98, 99, 102, 103,

138Future studies, 437, 439

g-factor, 142, 143, 160, 190, 191gas to dust ratio, 94gas-phase chemistry, 194grain Sizes, 268grain temperature, 271Greenstein effect, 105

Holn-London factors, 62, 63Hale-Bopp, 435Halley, 1, 2, 5–7, 16, 20–22, 24–29,

76, 77, 83, 85, 87, 88, 154, 157, 158,161, 163–168, 170, 174, 175,179–182, 191, 194, 198, 200–202,204, 207, 209

halos, 16, 20Haser’s model, 145, 148, 150, 151,

155, 159, 176, 190heating efficiency, 199heavy elements, 128, 183heliocentric variation, 154, 158, 162,

163, 175heteronuclear molecule, 56homonuclear molecule, 56hot band, 83, 159Hyakutake, 82, 85, 88, 115, 116, 329,

435hydrodynamic flow, 199hydrogen lines, 13, 16, 21

ice band, 290icy-conglomerate model, 24Impact of outside bodies, 424, 433in situ measurements, 23, 194, 281Infrared band passes, 269infrared lines, 85


Index 443

infrared observations, 253, 270, 272,283, 289–291, 298, 301

interplanetary dust particles, 258, 416interplanetary magnetic fields, 309Interstellar molecules, 428ion tail, 18, 22, 29, 303, 304, 309, 310,

323–325, 327, 335, 336ion-molecule reactions, 192isotopic abundances, 130isotopic shift, 58

Kramers-Kroning relation, 244Kuiper belt, 30, 390, 391, 393,

395–397

Lambda doubling, 64lifetime of the molecule, 142, 152linear polarization, 232, 237, 261,

264–266, 300long period comets, 7, 29, 383, 389,

390, 396, 398Lyman α isophotes, 186, 189Lyman α line of hydrogen, 16, 152

magnetic sector boundaries, 73, 324mass loss, 82, 360, 367, 377mass of grains, 272mass spectrometer, 88, 194, 196, 201material strength, 362, 363Maxwell-Garnett rule, 245meteor showers, 411, 413–415, 417,

422meteorites, 401, 406–411, 413, 416,

418, 424, 425Mie Theory, 228, 235mineralogy of dust, 28, 281minor constituents, 166, 178, 207missions, 22molecular bands, 77Molecular spectroscopy, 54molecules, 141–144, 146, 147, 151,

153, 155, 160, 162–164, 166, 167,169–177, 181–184, 187, 188, 192,199, 201, 203

Monte Carlo Model, 189, 201Monte-Carlo approach, 188, 210

Morphology, 339Morse potential, 63multi-component models, 297

new comets, 9, 395, 396non-gravitational force, 354, 356, 360,

369, 370, 375–378nucleus, 77, 82, 87, 90, 105, 107, 108,

115, 118, 339–346nucleus composition, 359

observed species, 27, 89Oort cloud, 9, 29, 30, 383–388, 390,

391optical constants, 242–246orbital elements, 41, 43origin, 4, 8, 10, 29, 30, 311, 335, 381,

391, 392, 395, 435, 438, 439ortho-to-para ratio, 371, 372, 376oscillations, 19, 24, 265, 319–321, 323,

325outbursts, 16, 348oxygen line, 121, 122, 180

parent molecules, 27, 146, 187, 188,202–204, 206, 207, 210

parent-daughter hypothesis, 146particle collection, 416, 432phase function, 231, 236, 237, 256,

257, 297photochemistry, 203photochemistry of water, 65photodissociation, 143photoionization rates, 143photometric theory, 151Planck’s law, 47plasma tail, 18–20, 308, 325, 328, 330polarization, 227, 232, 239, 240, 245,

261–268, 297, 300Polycyclic Aromatic Hydrocarbons,

69, 85, 286polyoxymethylene, 198presolar grains, 9, 409primordial material, 401, 419production rate, 141, 142, 144, 147,

148, 150–167, 169, 170, 174–177,



179, 182–184, 186–189, 191, 192,199, 204, 206, 207

Prompt emission, 79, 114, 159

radial velocity, 37, 45, 50radiation pressure, 214, 225, 227, 229,

230, 254, 269, 294–297, 299radicals, 87, 88, 90radio lines, 65, 118Rayleigh, 186reddening, 252–255relative abundances, 163, 164, 174,

183RKR potential, 63rotation, 339, 345, 356–358rotation of the nucleus, 25rotational structure, 98, 114, 132rotational temperature, 126, 128rotational transitions, 86

sand bank model, 363scale length, 146, 162, 178, 179, 188,

203Scattered intensity, 231Scattering functions, 247scattering theory, 228sector boundary, 324, 329, 330Shoemaker-Levy 9, 2, 14, 16, 17short period comets, 381, 389–391,

395, 398silicate feature, 276, 279silicate structure, 67size, 339, 340, 346, 351, 354, 375, 407,

412, 414, 416, 424Sodium Gas Tails, 222solar constant, 70solar radiation, 9, 70, 95, 96, 99–103,

105, 114, 129solar wind, 72–74, 303–315, 317–320,

322, 323, 325–328, 331–333solar wind interaction, 308, 309, 314space mission, 19spacecrafts to comets, 25spectra, 75–79, 81–83, 85–88, 90–93spectral features, 249, 278, 281spectroscopy, 47

splitting, 14, 15stardust, 22, 31, 287statistical equilibrium, 98, 101, 103,

105, 111, 114, 137streamers, 310, 324, 325structure, 24, 29, 56, 57, 63, 67, 68,

72, 73, 96, 97, 110, 113sun-grazing comets, 7, 8, 27, 77, 89surface brightness, 144, 145, 147, 150,

170, 203, 207, 208, 253–255surface gravity, 353Swan band sequence, 180Swings effect, 96, 119, 121, 155Syndyname, 186, 187, 214Syndyne, 214, 218, 222synthetic profile, 102, 108, 114

tails, 9, 10, 17, 18, 22, 323–326, 328,331, 336

Taxonomy, 396Tempel 1, 22, 25, 28, 29, 85, 174, 284,

289, 290, 355, 375temperature, 270, 271, 273, 274, 277,

279, 282, 283, 285, 290, 298, 392,396

terminal speed, 216, 225terrestrial water, 423theory of vapourization, 340tidal force, 17time evolution, 108Tunguska event, 2, 424turbulence, 303, 320, 322, 335

ultraviolet, 21, 22, 79, 81, 143, 144,155, 156, 167, 181, 183, 204, 205

ultraviolet coma, 26

vapourization theory, 344velocities in the coma, 155vibrational structure, 101vibrational temperature, 109, 111,

128vibrational transitions, 61visible coma, 82

waves, 303, 314, 319–325, 335


Index 445

Wild 2, 22, 25, 28, 29, 31, 87, 287,292, 355

X-rays, 330, 437

physics of comets 3rd ed. - k. swamy

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